This is the sixth post in the Infinity Series, the first of which is found here. For all posts, see the Infinity Series Portal.
In order to further explore the concept of infinity, one must temporarily move away from the concept of the cardinal number, and consider yet another type of number: the ordinal. The system of ordinal numbers is again entwined in set theory, and, unlike cardinal numbers, they can be expressed as sets. It is also important to know that any cardinal number can alternatively describe a family of sets, i.e. all of the sets that have that cardinality. For example, the cardinal number 3 not only describes a number of members that a set can have, but also represents the family of all sets with three members.
Formally, ordinal numbers are very similar to cardinal numbers in that, in finite situations, they represent properties of classes of sets. The property that an ordinal number represents is known as order type. However, the class of sets whose order type is described by an ordinal number much more specific than that of a cardinal number. To understand the type of set that has an order type can represent, one must first understand the concept of a well-ordered set.
A well-ordered set is a set that can be well-ordered by a relation. A relation, in this case, is just a quantity that connects pairs of elements. A good example is the relation "<" or "less than". To understand well-ordering, observe the example below.
Consider the set S = {2,5,7}, and the relation R = <
If "<" is restricted to the set S, it simply means that the two elements of each ordered pair of the relation must come from S, i.e. chosen from 2, 5, or 7. Of the six possible ordered pairs, (2,5), (2,7), (5,7), (5,2), (7,2), and (7,5), three are members of the relation <, namely (2,5), (2,7), and (5,7). This is because 2<5, 2<7, and 5<7. Obviously, the remaining pairs are not, as 5 is not less than 2, and so on.
In this case, "<" well-orders S, meaning that each (non-empty) subset of S has a "minimal" element. For the relation "<", the "minimal" element of a subset is simply its smallest member. Since it is easy to find the smallest number of any subset of S, it is a well-ordered set.
In contrast, consider the set of all whole numbers N = {1,2,3...}. The above relation "<" does well-order the set, because, for any subset, one can state the smallest element. However, this does not hold with the operation "greater than" or ">". For example, if the subset N' is the set of natural numbers greater than 3, i.e. {4,5,6...}, then there is no "minimal" element under the relation ">", as there is no element in the set that is greater than all the others. However, overall, the set is still "well-ordered", because there is at least one relation that does satisfy the needed conditions.
Now, returning to the class of sets a certain order type, consider all the well-ordered sets that can be put into one-to-one correspondence (i.e. all the well-ordered sets with a given cardinality). However, this correspondence is of a rather special type, matching up the "minimal" elements with each other, followed by the "next-to-minimal" elements, and so on. Every class of sets for which all the members are connected is an equivalence class, and all sets in this class have an identical order type, which is a certain ordinal number.
For instance, the above set S = {2,5,7} under "<", and the set T = {b,c,d} under the relation that will be defined as "alphabet" (which puts alphabet in order), can be related by a correspondence. The minimal element of set S under "<" is clearly 2, and the minimum of T under "alphabet" is b, with b being the letter in the set closest to the beginning of the alphabet. Since the other elements can be paired in a similar way, these two sets are therefore members of an equivalence class and are of the same order type (intuitively, they are this order type because they each have three elements).
However, for each equivalence class, it is convenient to chose one, and only one set to be representative of the entire class, and thus, the ordinal. The first such set is the empty set, Ø, having no elements and being the only member of its equivalence class. We identify this set with the ordinal number "0". Therefore,
0 = Ø = {}
Hence forth, the representative set of each equivalence class is defined as the set containing all of the previous representative sets. For example, the successor of 0 is denoted "1", or the second ordinal number, and is defined as
1 = {Ø}
Note that this set is not the empty set, but rather the set containing only the empty set. It is obvious that this set can be well-ordered, as, trivially, there is only one element to order. It is equally clear that this set is representative of the class of sets with a single element and that it can be put in a one-to-one correspondence with each set in that class. Continuing the pattern of each representative set containing all of those previously, we have
2 = {Ø, {Ø}},
3 = {Ø, {Ø}, {Ø, {Ø}}},
and so on. Note that "2", has 0 and 1 as members, and "3" has 0, 1, and 2 as members.
Henceforth, "ordinal number" will refer to a set of this type, and 0, 1, 2, 3,... will refer to ordinal numbers unless otherwise specified. These ordinal numbers are well-ordered by the membership relation, or the relation set up between two sets with the second containing the first. In other words, each element of an ordinal number set is a member of all subsequent elements in the set. For example,
4 = {Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}}}
In this set, each member is a member of each subsequent member, namely, Ø, being the first element of 4, is a member of {Ø}, {Ø, {Ø}}, and {Ø}, {Ø, {Ø}}}, which, of course, are just the ordinal numbers 1, 2, and 3. The minimal element for each ordinal number is Ø.
It is clear that one can go on to define any natural number as a set of this type, namely as the set of all previous ordinals. As with cardinal numbers, the first infinite ordinal number corresponds to the set of all natural numbers. This ordinal is denoted ω, and has every (ordinal) natural number as a member. It is clear that this set can be put in a one-to-one correspondence with the set of "regular" natural numbers, and it therefore has a cardinality of aleph-zero. However, the world of infinite ordinals is very different than that of infinite cardinals.
One crucial difference is the idea of succession in ordinals. From any ordinal α (greek letters are often used for ordinal variables), there is a successor to α, denoted α', which is defined as the next ordinal after α. Formally, α' is the smallest ordinal which has α as a member. Furthermore, α' can be constructed using the formula
α' = α ∪ {α}
where ∪ is the symbol for a union of sets. The concept of union entails combining all the members of two sets into one, and deleting any duplicates. For instance,
{1,4,6} ∪ {3,6,13} = {1,4,6,3,6,13} => {1,4,6,3,13}
Returning to the succession formula above, note the distinction between α and {α}. The first is an ordinal, but the second is not, being rather a set containing a single ordinal, α, as its only member. To more easily visualize this process, consider the simplest example:
0 = Ø
0' = Ø ∪ {Ø}
Now, the union of the empty set with any other set is simply the other set, as the empty set contributes no members to the result. Therefore,
0' = Ø ∪ {Ø} = {Ø},
the final term of which is, in fact, the ordinal 1! Though ordinal addition has not been defined, we can observe that the succession operation, in a more general case, identifies with the process of adding 1 (0+1=1 and 0'=1, etc.). The role that succession plays in infinite ordinals, as well as in operations on ordinals, is considered in the next post.
Sources: Axiomatic Set Theory by Patrick Suppes
Saturday, February 18, 2012
Friday, February 10, 2012
Infinity: Uncountable Sets
This is the fifth post of the Infinity Series, beginning here. For all posts, see the Infinity Series Portal.
In previous posts, the idea of an "uncountable set" has arisen, namely a set that has a cardinality greater than that of the natural numbers. We have already revealed that the sets of real numbers and complex numbers are uncountable, specifically having a cardinality that is two to the power of aleph-zero, the cardinality of the natural numbers. However, there are other sets which have this cardinality, as well as sets with one greater than this.
First, sets with the cardinality of the continuum will be listed. The sets of real and complex numbers fit into this category, along with any interval of either of them. For example, the set of real numbers between 1 and 5 (inclusive or exclusive) will have the cardinality of the continuum.
A somewhat more interesting case is the Cantor set, a famous entity in mathematics. It is obtained geometrically by beginning with a solid line segment, (mathematically, this segment is the real numbers on [0,1]) and on each step removing the middle third of every existing line segment. This process is illustrated below.
The first seven steps in producing the Cantor Set. This process is repeated infinitely many times. It appears that the entire original line segment will eventually be removed after an infinite number of the above steps, and this is, in a way, true. During the first step 1/3 of the total bar is removed, followed by 1/9 of the total for each remaining segment on the second step, then 1/27 of the total for each remaining part on the third step, and so on. The sum of these removals is 1/3+2/9+4/27+... since there are 2^n segments on the nth step (the amount removed on second step equals 2*1/9=2/9, and on the third, 4*1/27=4/27). The above sequence sums, in the limit as n increases without bound, to 1. In other words, 100% of the original segment is taken away as the Cantor Set is constructed.
Despite the apparent paradox, it is clear that for any segment remaining at any step of the above process, it will have two endpoints that are untouched even after an infinite number of steps. This is because only the middle of all such segments are removed. The remaining points, which all become endpoints at some step, are the points of the Cantor Set. For example, the points 0 and 1 are never removed, along with 1/3, 2/3, 1/9, 2/9, 7/9, 8/9 and many others. The cardinality of this set can be discovered by expressing all its points in base-3 (ternary) decimal notation. In it, 1/3=.1=.02222..., 2/3=.2=.12222..., 1=.22222..., 7/9=.20222..., etc. Many of these numbers have two forms, i.e. 1/3=.1=.0222..., but it can be shown that the Cantor Set contains only points whose base-3 decimal representation has at least one form (out of the two) that consists of only 2's and 0's (a more full discussion is found here). If one took all of these decimals and changed all of the 2's into 1's, a set of binary sequences would be produced. The Cantor set has all base-3 points consisting of only 2's and 0's, but after the function, which is clearly a bijection, it is now the set of all binary sequences, which has previously been shown to have the cardinality of the continuum. The Cantor Set does as well.
Mathematically, this is remarkable. Even though the Cantor Set is an infinitesimal fraction of the line segment [0,1], it still has the cardinality of the continuum. Since no two points of the final Cantor Set are "adjacent" to each other, the Cantor Set is defined to have the property of being nowhere dense. Simply put, this means that any interval containing two points of the Cantor Set does not necessarily contain a third. In contrast, there are an infinite number of rational numbers between any two one could chose, and the set of rational numbers is therefore dense. The fact that such a "nowhere dense" set can have the cardinality of the continuum is also incredible, because all of the discrete sets (natural numbers, integers, etc.) that we have examined have not had this cardinality. These are clearly nowhere dense, as there are real numbers between any two natural numbers, or any two integers.
Another very interesting case is the cardinality of the set of continuous functions on the real numbers. To find this cardinality, it is useful to use a proof of trichotomous exclusion. Such a proof confirms that a set must have a cardinality greater than or equal to that of a certain set, and then proves that it must simultaneously satisfy the condition of having a cardinality less than or equal to that of the same set. Since one quantity cannot be less than and greater than another, the cardinality of the two sets must be equal.
To go about the above proof, it is useful to reevaluate the definition of a continuous function. The "local" definition is of most use here. It states that for any value x on a continuous function, there exists a sufficiently small ε such that f(x+ε) approximates f(x) to an arbitrary accuracy σ. In addition, as ε goes to 0, σ will as well, as point Q becomes a better and better approximation to point P. This is all summarized in the figure below.
Compare the above to the discontinuous function below, where the open circle on the curve represents a discontinuity, the true function value f(x) being located at point P. Even with ε miniscule, the approximation to f(x) never reaches the desired accuracy of σ.
Having defined what it means to be continuous, it is now possible to find the cardinality of the set of continuous functions. The key concept is that knowing the value of a continuous function on all rational numbers is enough to determine it uniquely, i.e. to find its value at any real number.
To see this, consider the value x from above to be a certain real number (for our purpose, make it irrational). This number can be approximated to an arbitrarily high accuracy with a rational number, e.g. π≈3.14159=314159/10000, etc. and therefore ε can be made as small as desired. In the limit as ε goes to 0, the approximation to f(x) becomes exact, and the function value is uniquely determined. Therefore, the knowledge of the value of a continuous function over the rational numbers is enough to determine it.
The main idea is that the above shows that the any specific continuous function needs no more than a countable set of real numbers to define it. The cardinality of the set of all continuous functions is therefore either equal to or less than that of the set of the countable sets of real numbers, or the power set of real numbers, (a power set is the set of all combinations of elements of a given set, discussed in an earlier post) whose cardinality is 2 to the power of the cardinality of all countable sets: aleph-zero. The set of continuous functions therefore has no more than the cardinality of the continuum.
To confirm that this is the cardinality, we must still prove that the set of continuous functions is uncountable. There is a very simple way of doing this. Take a subset of the continuous functions, the constant functions, for which f(x)=k, k being a real number. Clearly, if one defines a set of all constant functions, including every possibility of k, the set will be equivalent in cardinality to the real numbers: the cardinality of the continuum. However, the constant functions are a subset of all continuous functions, the true cardinality must be equal to or greater than this. Since we have already confirmed that it is no greater, and now it is known that the cardinality of the set is no less than that of the real numbers, it follows that the two must be equal.
In contrast, discontinuous functions cannot be pinned down without their value at each and every real number. For example, the discontinuous (piecewise) function that defines f(x) to be equal to 1 for all x except π, and at π for the value to be 12, no degree of accuracy in the rational numbers can confirm that f(π) will equal twelve. The best approximations will simply yield f(π)=1, as they "expect" the function to be continuous at π. More generally, some discontinuous function could potentially have a jump at any or every real number. Therefore, it takes a set of real numbers to uniquely determine such a function.
The total set of functions (including both continuous and discontinuous) on the real numbers is then the power set of the real numbers, with a cardinality greater than any yet discussed. It is 2 to the power of the cardinality of the continuum, which is also known as beth-two, the symbol of which is illustrated below.
This is the third member of what is called the beth sequence. Beth-zero is equal to aleph-zero, and each subsequent element of the sequence is defined recursively as 2 to the power of the previous one. Beth-one is equal to 2 to the power of aleph-zero, or the cardinality of the continuum, and so on. Equivalently, each element is the cardinality of the power set of the previous element. The distinctions between the aleph series and the beth series are revealed in a later post. The next post is on a new type of number.
Sources: http://en.wikipedia.org/wiki/Cantor_set, etc.
In previous posts, the idea of an "uncountable set" has arisen, namely a set that has a cardinality greater than that of the natural numbers. We have already revealed that the sets of real numbers and complex numbers are uncountable, specifically having a cardinality that is two to the power of aleph-zero, the cardinality of the natural numbers. However, there are other sets which have this cardinality, as well as sets with one greater than this.
First, sets with the cardinality of the continuum will be listed. The sets of real and complex numbers fit into this category, along with any interval of either of them. For example, the set of real numbers between 1 and 5 (inclusive or exclusive) will have the cardinality of the continuum.
A somewhat more interesting case is the Cantor set, a famous entity in mathematics. It is obtained geometrically by beginning with a solid line segment, (mathematically, this segment is the real numbers on [0,1]) and on each step removing the middle third of every existing line segment. This process is illustrated below.
The first seven steps in producing the Cantor Set. This process is repeated infinitely many times. It appears that the entire original line segment will eventually be removed after an infinite number of the above steps, and this is, in a way, true. During the first step 1/3 of the total bar is removed, followed by 1/9 of the total for each remaining segment on the second step, then 1/27 of the total for each remaining part on the third step, and so on. The sum of these removals is 1/3+2/9+4/27+... since there are 2^n segments on the nth step (the amount removed on second step equals 2*1/9=2/9, and on the third, 4*1/27=4/27). The above sequence sums, in the limit as n increases without bound, to 1. In other words, 100% of the original segment is taken away as the Cantor Set is constructed.
Despite the apparent paradox, it is clear that for any segment remaining at any step of the above process, it will have two endpoints that are untouched even after an infinite number of steps. This is because only the middle of all such segments are removed. The remaining points, which all become endpoints at some step, are the points of the Cantor Set. For example, the points 0 and 1 are never removed, along with 1/3, 2/3, 1/9, 2/9, 7/9, 8/9 and many others. The cardinality of this set can be discovered by expressing all its points in base-3 (ternary) decimal notation. In it, 1/3=.1=.02222..., 2/3=.2=.12222..., 1=.22222..., 7/9=.20222..., etc. Many of these numbers have two forms, i.e. 1/3=.1=.0222..., but it can be shown that the Cantor Set contains only points whose base-3 decimal representation has at least one form (out of the two) that consists of only 2's and 0's (a more full discussion is found here). If one took all of these decimals and changed all of the 2's into 1's, a set of binary sequences would be produced. The Cantor set has all base-3 points consisting of only 2's and 0's, but after the function, which is clearly a bijection, it is now the set of all binary sequences, which has previously been shown to have the cardinality of the continuum. The Cantor Set does as well.
Mathematically, this is remarkable. Even though the Cantor Set is an infinitesimal fraction of the line segment [0,1], it still has the cardinality of the continuum. Since no two points of the final Cantor Set are "adjacent" to each other, the Cantor Set is defined to have the property of being nowhere dense. Simply put, this means that any interval containing two points of the Cantor Set does not necessarily contain a third. In contrast, there are an infinite number of rational numbers between any two one could chose, and the set of rational numbers is therefore dense. The fact that such a "nowhere dense" set can have the cardinality of the continuum is also incredible, because all of the discrete sets (natural numbers, integers, etc.) that we have examined have not had this cardinality. These are clearly nowhere dense, as there are real numbers between any two natural numbers, or any two integers.
Another very interesting case is the cardinality of the set of continuous functions on the real numbers. To find this cardinality, it is useful to use a proof of trichotomous exclusion. Such a proof confirms that a set must have a cardinality greater than or equal to that of a certain set, and then proves that it must simultaneously satisfy the condition of having a cardinality less than or equal to that of the same set. Since one quantity cannot be less than and greater than another, the cardinality of the two sets must be equal.
To go about the above proof, it is useful to reevaluate the definition of a continuous function. The "local" definition is of most use here. It states that for any value x on a continuous function, there exists a sufficiently small ε such that f(x+ε) approximates f(x) to an arbitrary accuracy σ. In addition, as ε goes to 0, σ will as well, as point Q becomes a better and better approximation to point P. This is all summarized in the figure below.
Compare the above to the discontinuous function below, where the open circle on the curve represents a discontinuity, the true function value f(x) being located at point P. Even with ε miniscule, the approximation to f(x) never reaches the desired accuracy of σ.
Having defined what it means to be continuous, it is now possible to find the cardinality of the set of continuous functions. The key concept is that knowing the value of a continuous function on all rational numbers is enough to determine it uniquely, i.e. to find its value at any real number.
To see this, consider the value x from above to be a certain real number (for our purpose, make it irrational). This number can be approximated to an arbitrarily high accuracy with a rational number, e.g. π≈3.14159=314159/10000, etc. and therefore ε can be made as small as desired. In the limit as ε goes to 0, the approximation to f(x) becomes exact, and the function value is uniquely determined. Therefore, the knowledge of the value of a continuous function over the rational numbers is enough to determine it.
The main idea is that the above shows that the any specific continuous function needs no more than a countable set of real numbers to define it. The cardinality of the set of all continuous functions is therefore either equal to or less than that of the set of the countable sets of real numbers, or the power set of real numbers, (a power set is the set of all combinations of elements of a given set, discussed in an earlier post) whose cardinality is 2 to the power of the cardinality of all countable sets: aleph-zero. The set of continuous functions therefore has no more than the cardinality of the continuum.
To confirm that this is the cardinality, we must still prove that the set of continuous functions is uncountable. There is a very simple way of doing this. Take a subset of the continuous functions, the constant functions, for which f(x)=k, k being a real number. Clearly, if one defines a set of all constant functions, including every possibility of k, the set will be equivalent in cardinality to the real numbers: the cardinality of the continuum. However, the constant functions are a subset of all continuous functions, the true cardinality must be equal to or greater than this. Since we have already confirmed that it is no greater, and now it is known that the cardinality of the set is no less than that of the real numbers, it follows that the two must be equal.
In contrast, discontinuous functions cannot be pinned down without their value at each and every real number. For example, the discontinuous (piecewise) function that defines f(x) to be equal to 1 for all x except π, and at π for the value to be 12, no degree of accuracy in the rational numbers can confirm that f(π) will equal twelve. The best approximations will simply yield f(π)=1, as they "expect" the function to be continuous at π. More generally, some discontinuous function could potentially have a jump at any or every real number. Therefore, it takes a set of real numbers to uniquely determine such a function.
The total set of functions (including both continuous and discontinuous) on the real numbers is then the power set of the real numbers, with a cardinality greater than any yet discussed. It is 2 to the power of the cardinality of the continuum, which is also known as beth-two, the symbol of which is illustrated below.
This is the third member of what is called the beth sequence. Beth-zero is equal to aleph-zero, and each subsequent element of the sequence is defined recursively as 2 to the power of the previous one. Beth-one is equal to 2 to the power of aleph-zero, or the cardinality of the continuum, and so on. Equivalently, each element is the cardinality of the power set of the previous element. The distinctions between the aleph series and the beth series are revealed in a later post. The next post is on a new type of number.
Sources: http://en.wikipedia.org/wiki/Cantor_set, etc.
Thursday, February 2, 2012
Infinity: Operations on Cardinals
Before reading this post, make sure you have read the first three parts of the Infinity Series, the first of which is found here. For all posts, see the Infinity Series Portal.
Having found the mathematical relationship between aleph-zero and the cardinality of the continuum, one wonders if it is possible to perform other operations with infinity cardinals, and whether these equations create any numbers not yet discussed. To start, take a simple addition from cardinal arithmetic:
What exactly does this equation mean? We are asked to "add" two quantities, one of which is an infinite cardinal, and one is a simple number. That it is possible to evaluate the sum (1) follows from the fact that aleph-zero and 1 are both cardinal numbers. Since they are of the same number system, they are "compatible" in a way, and can be combined by the use of sets. We have already proved (1) in the first post of the series, but let us recap. It has been discussed that adding the element 0 to the set of natural numbers does not change its cardinality, since the function y=x-1 from one set to the other is a bijection. In set notation, with vertical lines representing cardinality: |{0}|+|{1,2,3...}|=|{0,1,2,3...}| Note that if one replaces the values in this equation with the actual cardinalities, one obtains the equation (1) above! The equation in simple sets that has the same meaning as a cardinal equation will henceforth be known as the Corresponding Set Equality (CSE).
The generalization of (1) for any natural number n is
This can also be transformed into a CSE, since any subset of integers also has cardinality aleph-zero. The the set {-n+1,-n+2,...-2,-1,0} clearly has n elements, and can be added to the natural numbers to form the CSE, namely: |{-n+1,-n+2,...-2,-1,0}|+|{1,2,3...}|=|{-n+1,-n+2,...-2,-1,0,1,2,3...}|. By evaluating the cardinalities on both sides, one obtains equation (2). It is easy to continue on to other arithmetic operations, such as multiplication:
This particular equation also has a CSE. First consider the set of integers. It is clear that it can be split into two components, both of which have a cardinality of aleph-zero, namely the whole numbers: {0,1,2,3...} and the negative integers: {...-3,-2,-1} However, when these two sets are combined, the set of integers results, which we already know has cardinality aleph-zero as well. The CSE here is |{0,1,2,3...}|+|{-1,-2,-3...}|=|{...-3,-2,-1,0,1,2,3...}|. All three of the cardinalities are evaluated as aleph-zero, and (3) results. Now we shall prove the general multiplication result
for any natural number n. We have already determined that the rational numbers, and any infinite subset of them have cardinality aleph-zero. Consider the set {a/n,1+(a/n),2+(a/n),...}. This is simply the set of natural numbers with the rational number a/n added to each term. For a constant n, one could consider creating a set for each integral value of a from 0, to n-1, inclusive. This produces n sets of cardinality aleph-zero.
An example will make this more clear. Consider the case with n=3. For a=0, the set produced is simply the whole numbers ({0/3,1+(0/3),2+(0/3)...}={0,1,2...}) For a=1, the set is {1/3,1+(1/3),2+(1/3)...} and for a=2, the set is {2/3,1+(2/3),2+(2/3)...}, all of which have cardinality aleph-zero. The sum of these sets is {0,1/3,2/3,1,1+(1/3),1+(2/3),2...}, or the set of all multiples of 1/3, which also has the same cardinality. We have just proved (4) for n=3. The more general CSE for (4) is |{0/n,1+(0/n),2+(0/n)...}|+|{1/n,1+(1/n),2+(1/n)...}|+
|{2/n,1+(2/n),2+(2/n)...}|+...+|{(n-1)/n,1+((n-1)/n),2+((n-1)/n)...}|=
|{0,1/n,2/n...,1,1+(1/n),1+(2/n),...}|, the final set being the set of multiples of 1/n.
This equation is tedious, but it simply is the division of the set of multiples of 1/n into n parts, all of which have cardinality of aleph-zero. Since the set of multiples on the right hand side of the equation as an equivalent cardinality, this proves (4). Moving on, even the multiplication of two infinite quantities is possible.
The CSE for this equation follows from the countability of the set of ordered pairs. The set of ordered pairs (x,y) with natural numbers x and y can be split into components by setting a value for x, for example 1, and letting y vary among the natural numbers. For each constant value of x, a set of cardinality aleph-zero is generated, and since there are aleph-zero choices for x, the above result (5) follows. In CSE form, |{(1,1),(1,2),(1,3)...}|+|{(2,1),(2,2),(2,3)...}|+...=[the cardinality of the set of ordered pairs]. Each quantity in the equality has value aleph-zero when evaluated, and since there are as many members on the left side, it follows that the cardinality of the natural numbers, when multiplied by itself, yields the same quantity. This result can too be generalized to any positive integer power:
Since the case n=2 makes use of ordered pairs, it is natural to assume that higher powers will involve the corresponding ordered n-tuplet. This is correct. There are aleph-zero possibilities for each element of an integral ordered n-tuplet, and each choice of element contributes an aleph-zero to the product. The end result is aleph-zero to the nth power, but since it has already been said that the cardinalities of the sets of ordered n-tuplets for finite n are all aleph-zero, the equality (6) is a direct result.
Summarizing the above, no additions, multiplications, or nth powers, when applied to aleph-zero, change its value. However, it has already been shown that two taken to the power of aleph-zero produces a different infinite cardinal, namely the cardinality of the continuum. But what about a general natural number n taken to the same power, or even aleph-zero taken to the power of itself?
Remarkably, we find that this quantity is equal to the cardinality of the continuum! This can be derived intuitively from the result (6). Since aleph-zero to the power of n is equal to the cardinality of the set of the ordered n-tuplets, one obtains (7) by letting n increase without bound to aleph-zero, at which point one obtains the cardinality of the set of infinite sequences, which has previously been shown to be greater than aleph-zero, and named the cardinality of the continuum. Any other n taken to the aleph-zero power is also equal to the cardinality of the continuum, as such quantities would clearly be greater than 2 to that power and less than the left side of (7). Since these both have the same value, those of the general case do as well. This is all summarized below.
The next post explores uncountable sets, namely stating what other sets besides the real or complex numbers have a cardinality equal to, or even greater than, the cardinality of the continuum.
Sources: http://en.wikipedia.org/wiki/Power_set
Having found the mathematical relationship between aleph-zero and the cardinality of the continuum, one wonders if it is possible to perform other operations with infinity cardinals, and whether these equations create any numbers not yet discussed. To start, take a simple addition from cardinal arithmetic:
What exactly does this equation mean? We are asked to "add" two quantities, one of which is an infinite cardinal, and one is a simple number. That it is possible to evaluate the sum (1) follows from the fact that aleph-zero and 1 are both cardinal numbers. Since they are of the same number system, they are "compatible" in a way, and can be combined by the use of sets. We have already proved (1) in the first post of the series, but let us recap. It has been discussed that adding the element 0 to the set of natural numbers does not change its cardinality, since the function y=x-1 from one set to the other is a bijection. In set notation, with vertical lines representing cardinality: |{0}|+|{1,2,3...}|=|{0,1,2,3...}| Note that if one replaces the values in this equation with the actual cardinalities, one obtains the equation (1) above! The equation in simple sets that has the same meaning as a cardinal equation will henceforth be known as the Corresponding Set Equality (CSE).
The generalization of (1) for any natural number n is
(2)
This can also be transformed into a CSE, since any subset of integers also has cardinality aleph-zero. The the set {-n+1,-n+2,...-2,-1,0} clearly has n elements, and can be added to the natural numbers to form the CSE, namely: |{-n+1,-n+2,...-2,-1,0}|+|{1,2,3...}|=|{-n+1,-n+2,...-2,-1,0,1,2,3...}|. By evaluating the cardinalities on both sides, one obtains equation (2). It is easy to continue on to other arithmetic operations, such as multiplication:
(3)
This particular equation also has a CSE. First consider the set of integers. It is clear that it can be split into two components, both of which have a cardinality of aleph-zero, namely the whole numbers: {0,1,2,3...} and the negative integers: {...-3,-2,-1} However, when these two sets are combined, the set of integers results, which we already know has cardinality aleph-zero as well. The CSE here is |{0,1,2,3...}|+|{-1,-2,-3...}|=|{...-3,-2,-1,0,1,2,3...}|. All three of the cardinalities are evaluated as aleph-zero, and (3) results. Now we shall prove the general multiplication result
(4)
for any natural number n. We have already determined that the rational numbers, and any infinite subset of them have cardinality aleph-zero. Consider the set {a/n,1+(a/n),2+(a/n),...}. This is simply the set of natural numbers with the rational number a/n added to each term. For a constant n, one could consider creating a set for each integral value of a from 0, to n-1, inclusive. This produces n sets of cardinality aleph-zero.
An example will make this more clear. Consider the case with n=3. For a=0, the set produced is simply the whole numbers ({0/3,1+(0/3),2+(0/3)...}={0,1,2...}) For a=1, the set is {1/3,1+(1/3),2+(1/3)...} and for a=2, the set is {2/3,1+(2/3),2+(2/3)...}, all of which have cardinality aleph-zero. The sum of these sets is {0,1/3,2/3,1,1+(1/3),1+(2/3),2...}, or the set of all multiples of 1/3, which also has the same cardinality. We have just proved (4) for n=3. The more general CSE for (4) is |{0/n,1+(0/n),2+(0/n)...}|+|{1/n,1+(1/n),2+(1/n)...}|+
|{2/n,1+(2/n),2+(2/n)...}|+...+|{(n-1)/n,1+((n-1)/n),2+((n-1)/n)...}|=
|{0,1/n,2/n...,1,1+(1/n),1+(2/n),...}|, the final set being the set of multiples of 1/n.
This equation is tedious, but it simply is the division of the set of multiples of 1/n into n parts, all of which have cardinality of aleph-zero. Since the set of multiples on the right hand side of the equation as an equivalent cardinality, this proves (4). Moving on, even the multiplication of two infinite quantities is possible.
(5)
The CSE for this equation follows from the countability of the set of ordered pairs. The set of ordered pairs (x,y) with natural numbers x and y can be split into components by setting a value for x, for example 1, and letting y vary among the natural numbers. For each constant value of x, a set of cardinality aleph-zero is generated, and since there are aleph-zero choices for x, the above result (5) follows. In CSE form, |{(1,1),(1,2),(1,3)...}|+|{(2,1),(2,2),(2,3)...}|+...=[the cardinality of the set of ordered pairs]. Each quantity in the equality has value aleph-zero when evaluated, and since there are as many members on the left side, it follows that the cardinality of the natural numbers, when multiplied by itself, yields the same quantity. This result can too be generalized to any positive integer power:
(6)
Since the case n=2 makes use of ordered pairs, it is natural to assume that higher powers will involve the corresponding ordered n-tuplet. This is correct. There are aleph-zero possibilities for each element of an integral ordered n-tuplet, and each choice of element contributes an aleph-zero to the product. The end result is aleph-zero to the nth power, but since it has already been said that the cardinalities of the sets of ordered n-tuplets for finite n are all aleph-zero, the equality (6) is a direct result.
Summarizing the above, no additions, multiplications, or nth powers, when applied to aleph-zero, change its value. However, it has already been shown that two taken to the power of aleph-zero produces a different infinite cardinal, namely the cardinality of the continuum. But what about a general natural number n taken to the same power, or even aleph-zero taken to the power of itself?
(7)
Remarkably, we find that this quantity is equal to the cardinality of the continuum! This can be derived intuitively from the result (6). Since aleph-zero to the power of n is equal to the cardinality of the set of the ordered n-tuplets, one obtains (7) by letting n increase without bound to aleph-zero, at which point one obtains the cardinality of the set of infinite sequences, which has previously been shown to be greater than aleph-zero, and named the cardinality of the continuum. Any other n taken to the aleph-zero power is also equal to the cardinality of the continuum, as such quantities would clearly be greater than 2 to that power and less than the left side of (7). Since these both have the same value, those of the general case do as well. This is all summarized below.
The next post explores uncountable sets, namely stating what other sets besides the real or complex numbers have a cardinality equal to, or even greater than, the cardinality of the continuum.
Sources: http://en.wikipedia.org/wiki/Power_set
Wednesday, January 25, 2012
Infinity: The Cardinality of the Continuum
Before reading this post, make sure you have read Infinity: The First Transfinite Cardinal, and Infinity: Countable Sets.
In the previous posts of this series, it was established that the sets of natural numbers, integers, rational numbers, and even algebraic numbers have an equivalent cardinality: aleph-zero. However, not all real numbers fall under the umbrella of algebraic numbers. All of the numbers that are real but non-algebraic are irrational, and are specifically known as transcendental. Numbers such as e and π are transcendental.
To determine the cardinality of the real numbers, this problem can be again simplified to a problem involving ordered n-tuplets. This is done by considering the construction of an arbitrary real number. The general real number has a finite whole number part, followed by an infinite decimal expansion. For example, the real number π has a whole number part of 3, and a decimal expansion of .14159265... It is simpler to just ignore the whole number part, and focus on the real numbers on the interval (0,1). All of these are defined uniquely (almost, as we will see below) by their infinite decimal expansion. Therefore, each of these numbers is defined by an ordered n-tuplet, with n being infinite, and of the form
(a1,a2,a3...)
Since all values in this sequence are place values, each must be 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. However, no clarity is lost if these real numbers are converted to binary, and each number still as a unique infinite decimal expansion, this time only incorporating 1's and 0's. We have the just simplified the problem to determining whether the set of all infinite sequences consisting of 1's and 0's is countable, i.e. whether it has a cardinality of aleph-zero.
Georg Cantor was the first to devise this method, and through infinite binary sequences found a very elegant way to find the cardinality of the real numbers, through proof by contradiction. It is called the diagonal argument. To understand this argument, consider all possible infinite binary sequences as making up a set, called S. Each element Sn is then an infinite binary sequence. If the cardinality of the real numbers is aleph-zero, then each element in the set can be numbered Sn, with n being a natural number. The first few elements of the set are shown below:
The actual ordering of this set is arbitrary, since if the cardinality of S is aleph-zero, all sequences with be covered eventually. For the next part of the proof, consider a sequence Sx, which is constructed by taking the nth element of each Sn and reversing it, i.e. 0 becomes 1, and 1 becomes 0. This is illustrated below:
The nth element of every nth sequence is bolded, and for each bolded element, the opposite one is placed in the sequence Sx. The resulting sequence is clearly an infinite binary sequence, and therefore is a member of the set S. However, by definition, it is different from any sequence in the set (S1,S2,S3...) because for any sequence Sn, with n a natural number, the nth element of the sequence is different from that of Sx. Therefore, assigning a natural number to each infinite binary sequence does not cover all such sequences, and the function from the natural numbers to S is not a bijection. Therefore, the set S has a cardinality greater then aleph-zero. Sets with cardinalities greater than aleph-zero are also known as uncountable.
When extending this back to our real number problem, there are a few slight glitches in this system, one of which being that an infinite binary decimal expansion such as .001111... with infinite 1's is actually equal to another, namely .010000... Therefore, each real number does not quite have a unique decimal expansion. However, this problem can be resolved.
Only numbers with terminating decimal expansions can be expressed in the two ways shown above, and in binary, the only such numbers are those whose denominators involve only a power of 2. For example, 1/2=.1000...=.0111..., and 5/8=.101000...=.100111... These numbers can be collected into a set of their own, called A, the first few members of which are {1/2, 1/4, 3/4, 1/8, 3/8,...}.
This is a subset of the rational numbers, and therefore has cardinality aleph-zero. The set of infinite binary sequences with infinite 0's or infinite 1's (which starts {.1000...,.0111...,.01000...,.00111...,...}) merely has two elements for each element of set A, and still has a cardinality of aleph-zero. (just as the sets {1,2,3...} and {1,-1,2,-2...} have the same cardinality, even though there are two elements in the latter whose absolute values correspond to the former) Because of this, a bijection can be set up between them, corresponding .1000... to 1/2, .0111... to 1/4, and so on. Adding this to the original set S, one finds that each real number on (0,1) now corresponds to a single unique infinite binary sequence.
Clearly, the cardinality of the entire set of real numbers must be greater than or equal to that of the real numbers on (0,1), and so we have now proved that
The cardinality of the real numbers, known as the cardinality of the continuum and denoted by
, is strictly greater then the cardinality of the natural numbers, aleph-zero. In other words
> ℵ0
More number systems still remain, including the complex numbers. However, it is fairly easy to see that the complex numbers still have the cardinality of the continuum, as any complex number can be defined uniquely as an ordered pair of two real numbers (a,b). Also, through a similar method that was used for integral ordered n-tuplets that is not detailed here, it can be proved that sets of ordered n-tuplets of real numbers or of complex numbers both have the cardinality of the continuum. This result also seems intuitively correct from previous examinations of ordered n-tuplets.
Using Cantor's diagonal argument, it was established that the cardinality of the set of real numbers was greater than that of natural numbers, in other words that there is no bijection between them. However, it has not been found exactly what this cardinality of the continuum actually is, or how to relate these quantities to each other in any way.
In order to accomplish this, one must first understand the concept of a power set. A power set is denoted P(S), where S is any ordinary set. Every set has a corresponding power set, and the above statement says that the power set of S is called P(S).
To construct the power set for any set S, take each unique combination of the elements in S, and put it into its own set. Then compile all of these sets and place them within another set. This is the power set, P(S). For example, the set {1,2,3} includes eight unique combinations of the elements contained within it, namely: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, and {1,2,3}. (the first of these is included as a choice of zero elements from the set) All of these are then included in a larger set, yielding {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. In conclusion:
P({1,2,3})={{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
If one compares the cardinality of these two sets, where each subset of the latter is a single element, it is easy to see that they are 3 and 8, respectively. One also notices that 2^3=8. This is no coincidence. The total number of combinations of n elements is (2^n)-1, and when one adds the empty set to this total, 2^n. Expressed in formula form, with the cardinality of a set S written as |S|:
Applying this knowledge to what we know about the set of natural numbers, it is easy to identify the set of all real numbers as the set of combinations of natural numbers, using the technique of binary sequences that was used previously. Combinations of natural numbers can be interpreted at the decimal expansion of real numbers through binary sets. For example, one can draw the combination {1,4,7} from the set of natural numbers {1,2,3...} and take it to mean .1100111000... which is a decimal sequence in which the binary expansions of 1 4, and 7 are joined, and then followed by zeros. There are also infinite subsets of the natural numbers, such as the even numbers {2,4,6...}, which will provide an infinite decimal expansion. This is not a precise mathematical way of doing this, but it serves as an intuitive glimpse into the cardinality of the continuum.
One can also choose the first element of the subset to represent the whole number part, which is always a finite natural number for all positive reals. Using this method, every real number can be generated from a subset (finite or infinite) of the natural numbers, and the real set is the power set of the natural numbers. This proves that
In other words, the cardinality of the continuum is two to the power of aleph-zero. But how can one even comprehend arithmetic operations with cardinal numbers, and infinite ones at that? What other properties does aleph-zero have? The answers can be defined through sets, and are discussed in the next post.
Sources: http://en.wikipedia.org/wiki/Cardinality_of_the_continuum, http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
In the previous posts of this series, it was established that the sets of natural numbers, integers, rational numbers, and even algebraic numbers have an equivalent cardinality: aleph-zero. However, not all real numbers fall under the umbrella of algebraic numbers. All of the numbers that are real but non-algebraic are irrational, and are specifically known as transcendental. Numbers such as e and π are transcendental.
To determine the cardinality of the real numbers, this problem can be again simplified to a problem involving ordered n-tuplets. This is done by considering the construction of an arbitrary real number. The general real number has a finite whole number part, followed by an infinite decimal expansion. For example, the real number π has a whole number part of 3, and a decimal expansion of .14159265... It is simpler to just ignore the whole number part, and focus on the real numbers on the interval (0,1). All of these are defined uniquely (almost, as we will see below) by their infinite decimal expansion. Therefore, each of these numbers is defined by an ordered n-tuplet, with n being infinite, and of the form
(a1,a2,a3...)
Since all values in this sequence are place values, each must be 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. However, no clarity is lost if these real numbers are converted to binary, and each number still as a unique infinite decimal expansion, this time only incorporating 1's and 0's. We have the just simplified the problem to determining whether the set of all infinite sequences consisting of 1's and 0's is countable, i.e. whether it has a cardinality of aleph-zero.
Georg Cantor was the first to devise this method, and through infinite binary sequences found a very elegant way to find the cardinality of the real numbers, through proof by contradiction. It is called the diagonal argument. To understand this argument, consider all possible infinite binary sequences as making up a set, called S. Each element Sn is then an infinite binary sequence. If the cardinality of the real numbers is aleph-zero, then each element in the set can be numbered Sn, with n being a natural number. The first few elements of the set are shown below:
The actual ordering of this set is arbitrary, since if the cardinality of S is aleph-zero, all sequences with be covered eventually. For the next part of the proof, consider a sequence Sx, which is constructed by taking the nth element of each Sn and reversing it, i.e. 0 becomes 1, and 1 becomes 0. This is illustrated below:
The nth element of every nth sequence is bolded, and for each bolded element, the opposite one is placed in the sequence Sx. The resulting sequence is clearly an infinite binary sequence, and therefore is a member of the set S. However, by definition, it is different from any sequence in the set (S1,S2,S3...) because for any sequence Sn, with n a natural number, the nth element of the sequence is different from that of Sx. Therefore, assigning a natural number to each infinite binary sequence does not cover all such sequences, and the function from the natural numbers to S is not a bijection. Therefore, the set S has a cardinality greater then aleph-zero. Sets with cardinalities greater than aleph-zero are also known as uncountable.
When extending this back to our real number problem, there are a few slight glitches in this system, one of which being that an infinite binary decimal expansion such as .001111... with infinite 1's is actually equal to another, namely .010000... Therefore, each real number does not quite have a unique decimal expansion. However, this problem can be resolved.
Only numbers with terminating decimal expansions can be expressed in the two ways shown above, and in binary, the only such numbers are those whose denominators involve only a power of 2. For example, 1/2=.1000...=.0111..., and 5/8=.101000...=.100111... These numbers can be collected into a set of their own, called A, the first few members of which are {1/2, 1/4, 3/4, 1/8, 3/8,...}.
This is a subset of the rational numbers, and therefore has cardinality aleph-zero. The set of infinite binary sequences with infinite 0's or infinite 1's (which starts {.1000...,.0111...,.01000...,.00111...,...}) merely has two elements for each element of set A, and still has a cardinality of aleph-zero. (just as the sets {1,2,3...} and {1,-1,2,-2...} have the same cardinality, even though there are two elements in the latter whose absolute values correspond to the former) Because of this, a bijection can be set up between them, corresponding .1000... to 1/2, .0111... to 1/4, and so on. Adding this to the original set S, one finds that each real number on (0,1) now corresponds to a single unique infinite binary sequence.
Clearly, the cardinality of the entire set of real numbers must be greater than or equal to that of the real numbers on (0,1), and so we have now proved that
The cardinality of the real numbers, known as the cardinality of the continuum and denoted by
, is strictly greater then the cardinality of the natural numbers, aleph-zero. In other words > ℵ0More number systems still remain, including the complex numbers. However, it is fairly easy to see that the complex numbers still have the cardinality of the continuum, as any complex number can be defined uniquely as an ordered pair of two real numbers (a,b). Also, through a similar method that was used for integral ordered n-tuplets that is not detailed here, it can be proved that sets of ordered n-tuplets of real numbers or of complex numbers both have the cardinality of the continuum. This result also seems intuitively correct from previous examinations of ordered n-tuplets.
Using Cantor's diagonal argument, it was established that the cardinality of the set of real numbers was greater than that of natural numbers, in other words that there is no bijection between them. However, it has not been found exactly what this cardinality of the continuum actually is, or how to relate these quantities to each other in any way.
In order to accomplish this, one must first understand the concept of a power set. A power set is denoted P(S), where S is any ordinary set. Every set has a corresponding power set, and the above statement says that the power set of S is called P(S).
To construct the power set for any set S, take each unique combination of the elements in S, and put it into its own set. Then compile all of these sets and place them within another set. This is the power set, P(S). For example, the set {1,2,3} includes eight unique combinations of the elements contained within it, namely: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, and {1,2,3}. (the first of these is included as a choice of zero elements from the set) All of these are then included in a larger set, yielding {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. In conclusion:
P({1,2,3})={{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
If one compares the cardinality of these two sets, where each subset of the latter is a single element, it is easy to see that they are 3 and 8, respectively. One also notices that 2^3=8. This is no coincidence. The total number of combinations of n elements is (2^n)-1, and when one adds the empty set to this total, 2^n. Expressed in formula form, with the cardinality of a set S written as |S|:
Applying this knowledge to what we know about the set of natural numbers, it is easy to identify the set of all real numbers as the set of combinations of natural numbers, using the technique of binary sequences that was used previously. Combinations of natural numbers can be interpreted at the decimal expansion of real numbers through binary sets. For example, one can draw the combination {1,4,7} from the set of natural numbers {1,2,3...} and take it to mean .1100111000... which is a decimal sequence in which the binary expansions of 1 4, and 7 are joined, and then followed by zeros. There are also infinite subsets of the natural numbers, such as the even numbers {2,4,6...}, which will provide an infinite decimal expansion. This is not a precise mathematical way of doing this, but it serves as an intuitive glimpse into the cardinality of the continuum.
One can also choose the first element of the subset to represent the whole number part, which is always a finite natural number for all positive reals. Using this method, every real number can be generated from a subset (finite or infinite) of the natural numbers, and the real set is the power set of the natural numbers. This proves that
In other words, the cardinality of the continuum is two to the power of aleph-zero. But how can one even comprehend arithmetic operations with cardinal numbers, and infinite ones at that? What other properties does aleph-zero have? The answers can be defined through sets, and are discussed in the next post.
Sources: http://en.wikipedia.org/wiki/Cardinality_of_the_continuum, http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
Tuesday, January 17, 2012
Infinity: Countable Sets
Before reading this post, read Infinity: The First Transfinite Cardinal.
At the end of the previous post, it was stated that any infinite set or subset of a number system defined by integral ordered n-tuplets, where n is a natural number, has a cardinality of aleph-zero. This statement is more easily understood with examples.
We have already seen that natural numbers can be put in one-to-one correspondence with any set of n-tuplets that contains only integers. When n=1, the resulting number system is simply the integers themselves: {...-3,-2,-1,0,1,2,3...} Furthermore, the above guarantees that any infinite subset of integers will have equivalent cardinality. The even numbers {2,4,6...}, multiples of 10, {10,20,30...}, and even the powers of 2 {1,2,4,8,16...} all can be accessed from the natural numbers through a simple bijection (in this case the functions y=2x, y=10x, and y=2^x, respectively) and therefore have the same cardinality, aleph-zero.
However, the above theorem states that a number system defined by integral ordered n-tuplets for any finite n are also allowed. First, consider the ordered pairs. How does one define a number system with the general integral pair (a,b)? One way, is to view these numbers as the solutions to polynomial equations with coefficients a and b, namely
ax+b=0
Solving for x, one obtains x=-b/a. The general solutions of these equations are the rational numbers, any numbers that can be formed by the ratio of two integers. Admittedly, using the spiral method to pair each natural number with an ordered pair does not hold up well when converted into rational numbers. For instance, the ordered pairs (-2,1) and (-4,2) both produce the same solution, namely 1/2, and pairs such as (0,1) are not defined at all! Therefore, the function from natural numbers to rational numbers through the spiral method is not a bijection.
These problems can be resolved, however. One way is to simply discard duplicates and undefined ordered pairs. The natural numbers corresponding to unique rational numbers will henceforth be known as unique ordered pair numbers, or UOPN's for short. The first few are
2 -> (1,0) -> 0
3 -> (1,1) -> -1
5 -> (-1,1) -> 1
10 -> (2,-1) -> 1/2
12 -> (2,1) -> -1/2
14 -> (1,2) -> -2
In the above expression, the first number of each row is a natural number, followed by the corresponding ordered pair defined by the spiral rule, and the final number is the rational number that results using the polynomial method on the ordered pair. Since it is clear that there are an infinite number of rational numbers accessed by the above series, one can set up a bijection pairing each natural number n to the nth UOPN. This would change the set {1,2,3,4...} into {2,3,5,10...}. The UOPN's then have the same cardinality as the natural numbers, and therefore the rational numbers do as well.
The result just established is remarkable. Despite there being infinite rational numbers between the natural numbers 0 and 1 alone, the cardinality of both of these sets are identical. But this still isn't the end of it. The theorem also deals with numbers defined by ordered n-tuplets. Continuing the theme of using ordered n-tuplets to define integral polynomial equations, the solutions of the resulting equation give the value of the ordered n-tuplet. For example, the ordered quadruplet (1,0,0,-2) corresponds to
x^3+0x^2+0x-2=x^3-2=0,
the real solution of which is the cube root of two. However, with higher degree polynomials, there may be multiple solutions. Consider the polynomial graph below.
This is the graph of x^3-2x^2-x+1, the ordered quadruplet for which would be (1,-2,-1,1). In this case, there are three solutions to the polynomial equation x^3-2x^2-x+1=0 on the real number line, i.e. the three intersections of the graph with the x-axis. How then does one avoid ambiguity? Which of the three solutions does (1,-2,-1,1) represent? The solution lies in specifying an interval on which the solution is found. For example, the first solution, to the left of the origin, lies on the interval [-1,0], and with this constraint, the unique solution can be specified.
Generally, given a ordered n-tuplet of the form (A1,A2,A3...,An-4,B1,B2,C1,C2), one specifies the corresponding number to be a zero of the polynomial
(A1)x^(n-4)+(A2)x^(n-3)+...(An-3)x+An-4
on the interval whose endpoints are the rational numbers defined by the ordered pairs (B1,B2) and (C1,C2).
It is now clear that any solution of any polynomial equation of integral coefficients has an accompanying natural number. Since these solutions can be set up in a one-to-one correspondence with some subset of the natural numbers, one can be set up with the natural numbers themselves. As before, this implies an identical cardinality, i.e. the set of the solutions to integral polynomial equations has a cardinality of aleph-zero.
Just what are these numbers? We know that they are the general solutions of integral polynomial equations, but what form do they take? This specific set of numbers are called the algebraic numbers, and they include square roots, cube roots, and for that matter any nth root. They also include more complicated sets of nested roots, such as numbers of the form
for integer a, b, and c. Any number with nested roots such as this is algebraic. Specifically, algebraic numbers encompass all rational numbers along with many irrational numbers, but not all real numbers are algebraic. For example, π and e are not algebraic, and cannot be expressed as the solutions of polynomials of any finite degree.
All of the above sets have a cardinality the same as that of the natural numbers, and they are therefore denoted countable sets, named after the ability to count natural numbers. All of the rational numbers, and even some irrational numbers are countable, but one hurdle remains: the real numbers. All of the surprising discoveries above suggest that the idea of real numbers being countable is not an implausible notion. The answer is revealed in the next post.
Sources: http://en.wikipedia.org/wiki/Cardinal_number
At the end of the previous post, it was stated that any infinite set or subset of a number system defined by integral ordered n-tuplets, where n is a natural number, has a cardinality of aleph-zero. This statement is more easily understood with examples.
We have already seen that natural numbers can be put in one-to-one correspondence with any set of n-tuplets that contains only integers. When n=1, the resulting number system is simply the integers themselves: {...-3,-2,-1,0,1,2,3...} Furthermore, the above guarantees that any infinite subset of integers will have equivalent cardinality. The even numbers {2,4,6...}, multiples of 10, {10,20,30...}, and even the powers of 2 {1,2,4,8,16...} all can be accessed from the natural numbers through a simple bijection (in this case the functions y=2x, y=10x, and y=2^x, respectively) and therefore have the same cardinality, aleph-zero.
However, the above theorem states that a number system defined by integral ordered n-tuplets for any finite n are also allowed. First, consider the ordered pairs. How does one define a number system with the general integral pair (a,b)? One way, is to view these numbers as the solutions to polynomial equations with coefficients a and b, namely
ax+b=0
Solving for x, one obtains x=-b/a. The general solutions of these equations are the rational numbers, any numbers that can be formed by the ratio of two integers. Admittedly, using the spiral method to pair each natural number with an ordered pair does not hold up well when converted into rational numbers. For instance, the ordered pairs (-2,1) and (-4,2) both produce the same solution, namely 1/2, and pairs such as (0,1) are not defined at all! Therefore, the function from natural numbers to rational numbers through the spiral method is not a bijection.
These problems can be resolved, however. One way is to simply discard duplicates and undefined ordered pairs. The natural numbers corresponding to unique rational numbers will henceforth be known as unique ordered pair numbers, or UOPN's for short. The first few are
2 -> (1,0) -> 0
3 -> (1,1) -> -1
5 -> (-1,1) -> 1
10 -> (2,-1) -> 1/2
12 -> (2,1) -> -1/2
14 -> (1,2) -> -2
In the above expression, the first number of each row is a natural number, followed by the corresponding ordered pair defined by the spiral rule, and the final number is the rational number that results using the polynomial method on the ordered pair. Since it is clear that there are an infinite number of rational numbers accessed by the above series, one can set up a bijection pairing each natural number n to the nth UOPN. This would change the set {1,2,3,4...} into {2,3,5,10...}. The UOPN's then have the same cardinality as the natural numbers, and therefore the rational numbers do as well.
The result just established is remarkable. Despite there being infinite rational numbers between the natural numbers 0 and 1 alone, the cardinality of both of these sets are identical. But this still isn't the end of it. The theorem also deals with numbers defined by ordered n-tuplets. Continuing the theme of using ordered n-tuplets to define integral polynomial equations, the solutions of the resulting equation give the value of the ordered n-tuplet. For example, the ordered quadruplet (1,0,0,-2) corresponds to
x^3+0x^2+0x-2=x^3-2=0,
the real solution of which is the cube root of two. However, with higher degree polynomials, there may be multiple solutions. Consider the polynomial graph below.
This is the graph of x^3-2x^2-x+1, the ordered quadruplet for which would be (1,-2,-1,1). In this case, there are three solutions to the polynomial equation x^3-2x^2-x+1=0 on the real number line, i.e. the three intersections of the graph with the x-axis. How then does one avoid ambiguity? Which of the three solutions does (1,-2,-1,1) represent? The solution lies in specifying an interval on which the solution is found. For example, the first solution, to the left of the origin, lies on the interval [-1,0], and with this constraint, the unique solution can be specified.
Generally, given a ordered n-tuplet of the form (A1,A2,A3...,An-4,B1,B2,C1,C2), one specifies the corresponding number to be a zero of the polynomial
(A1)x^(n-4)+(A2)x^(n-3)+...(An-3)x+An-4
on the interval whose endpoints are the rational numbers defined by the ordered pairs (B1,B2) and (C1,C2).
It is now clear that any solution of any polynomial equation of integral coefficients has an accompanying natural number. Since these solutions can be set up in a one-to-one correspondence with some subset of the natural numbers, one can be set up with the natural numbers themselves. As before, this implies an identical cardinality, i.e. the set of the solutions to integral polynomial equations has a cardinality of aleph-zero.
Just what are these numbers? We know that they are the general solutions of integral polynomial equations, but what form do they take? This specific set of numbers are called the algebraic numbers, and they include square roots, cube roots, and for that matter any nth root. They also include more complicated sets of nested roots, such as numbers of the form
for integer a, b, and c. Any number with nested roots such as this is algebraic. Specifically, algebraic numbers encompass all rational numbers along with many irrational numbers, but not all real numbers are algebraic. For example, π and e are not algebraic, and cannot be expressed as the solutions of polynomials of any finite degree.
All of the above sets have a cardinality the same as that of the natural numbers, and they are therefore denoted countable sets, named after the ability to count natural numbers. All of the rational numbers, and even some irrational numbers are countable, but one hurdle remains: the real numbers. All of the surprising discoveries above suggest that the idea of real numbers being countable is not an implausible notion. The answer is revealed in the next post.
Sources: http://en.wikipedia.org/wiki/Cardinal_number
Monday, January 9, 2012
Infinity: The First Transfinite Cardinal
In mathematics, infinity, often denoted ∞, is defined as exceeding all natural numbers, or, conversely, as the limiting value as a variable n increases without bound. ∞ has always been regarded as a sort of mystical quantity, ever out of reach from most mathematical concepts and calculations. However, through set theory, insights into infinity, in fact multiple infinities, can be gained.
A set is a series of elements, such as {1,2,3} or {12,58,-1,4}. The number of elements in a set is known as its cardinality. For example, {1,2,3} has a cardinality of 3, and {12,58,-1,4} has a cardinality of 4. Any number that can represent the cardinality of a set is known as a cardinal number. 3 and 4, as demonstrated above, are examples of cardinal numbers.
In fact, every natural number 1, 2, 3... is a cardinal number, and even 0 is a cardinal number, as it measures the number of elements in the empty set {}, also written Ø.
Now, it is useful to define functions on sets, namely rules for changing one set into another. The most important of these are known as bijections, and they are defined as functions of sets that preserve the cardinality of a set.
The above diagram is a pictorial representation of a bijection, defined as a function that maps each point in set X to exact one point in set Y, in other words a one-to-one correspondence. It is clear that such a function, when applied to a set, will preserve its cardinality. For example, the function y=x-1 maps the set {2,3,4} to the set {1,2,3}, each of the sets having a cardinality of exactly 3. Since this function preserves the cardinality of all sets in the same matter, it is a bijection.
It is also possible to define an infinite set, or a set with an infinite number of elements. The simplest of these is the set of all natural numbers, namely {1,2,3,4...}. This set has infinite cardinality, and the cardinal number representing this set is the first so-called transfinite cardinal. It is denoted aleph-zero, or ℵ0.
Aleph-zero is not contained within the normal number system that we think of, but rather describes the size of the set containing all natural numbers. One could then wonder whether adding another element, for example 0, to the set would result in a cardinality of ℵ0+1. This intuitively seems reasonable, but it is not the case. It was earlier shown that the function y=x-1 is a bijection, so that, when applied to the above set X, will preserve its cardinality of aleph-zero. The result is as follows:
X={1,2,3,4...}
Y=X-1={0,1,2,3...}
Therefore, the set containing all whole numbers, which include both the natural numbers and 0, also has a cardinality of aleph-zero.
At first glance, the above result appears paradoxical. It seems that by subtracting all the terms by 1, an element is added to the beginning of the set, but taken off the end. This is certainly true for finite sets of natural numbers. For any finite natural number n, if
X={1,2,3...,n-1,n}, then
Y=X-1={0,1,2...,n-1}
However, as n increases without bound towards ∞, n-1 is ultimately indistinguishable from n, as ∞-1 is still ∞. In addition to this, when one considers any natural number in the infinite set {1,2,3,4...}, it is decreased by 1 when the function is applied, but there is always another number to take its place. When one takes these points into consideration, the result begins to make sense.
One can easily use the functions y=x-2, y=x-3... etc. to incorporate the elements -1, -2, etc. with the same logic as before, generalizing the above statement to include any finite number of negative integers.
This is only the beginning. Next, consider the function defined below.
If x is odd, then y=(x-1)/2
If x is even, then y=-x/2
The above is not a usual function that can be defined in simple operators of x. However, it is still a bijection when applied to the set of natural numbers, as each element of the set {1,2,3,4...} is transferred to a unique number in a second set. This set is {0,-1,1,-2,2-3,3...}. Remarkably, the output of this function covers all integers! Again, it is easy to see that for any integer one could choose, there is always a natural number that produces it with the above function. Therefore, the total set of integers still has the same cardinality, namely aleph-zero, as the natural numbers.
Nor does the fun stop there! Next consider the mapping of a set to a set of ordered pairs, namely assigning a set of two integers to each natural number. This can be done in the following way:
1 -> (0,0), 2 -> (1,0), 3 -> (1,1), 4 -> (0,1), 5 -> (-1,1), 6 -> (-1,0), 7 -> (-1,-1), 8 -> (0,-1), 9 -> (1,-1), 10 -> (2, -1)...
The exact pattern of these ordered pairs can take several different forms, the above being one of these. Initially, the above sequence seems to have no clear pattern, but it does have a clear geometric significance.
The above pattern lists all the integral ordered pairs (white circles) reached as one follows a rectangular counterclockwise spiral beginning at the origin (the black path). Clearly, by following this path for a sufficient distance, we will visit any ordered pair of integers (a,b) that we could choose!
Before exploring the implications of the above statement, we must make one more logical step. Consider an ordered triplet of the form (a,b,c), again with a, b, and c integers. One can use a similar system to the one above to set up a one-to-one correspondence between the natural numbers and these triplets. The first few terms are listed below:
1 -> (0,0,0), 2 -> (0,0,1), 3 -> (0,1,1), 4 -> (1,1,1), 5 -> (1,0,1), 6 -> (1,-1,1), 7 -> (0,-1,1), 8 -> (-1,-1,1), 9 -> (-1,0,1), 10 -> (-1,1,1)...
Just as before, this can be visualized as a rectangular spiral in the three dimensional coordinate system, where each point is given coordinates (x,y,z). In fact, this pattern, and its geometric interpretation, continue for any order n-tuplets, each consisting of elements (a1,a2,a3...an) and representing a spiral in the n-dimensional Cartesian system.
But what does this mean in terms of the number systems? Clearly, the set of all integral n-tuplets, for any finite n, has a cardinality of aleph-zero. From this, we draw the similar result that
Any infinite set or subset of a number system whose members can be represented by integral ordered n-tuplets, with n a natural number, has the same cardinality as the set of natural numbers, namely aleph-zero.
The implications of the above statement are explored in the next post.
Sources: http://en.wikipedia.org/wiki/Cardinal_number
A set is a series of elements, such as {1,2,3} or {12,58,-1,4}. The number of elements in a set is known as its cardinality. For example, {1,2,3} has a cardinality of 3, and {12,58,-1,4} has a cardinality of 4. Any number that can represent the cardinality of a set is known as a cardinal number. 3 and 4, as demonstrated above, are examples of cardinal numbers.
In fact, every natural number 1, 2, 3... is a cardinal number, and even 0 is a cardinal number, as it measures the number of elements in the empty set {}, also written Ø.
Now, it is useful to define functions on sets, namely rules for changing one set into another. The most important of these are known as bijections, and they are defined as functions of sets that preserve the cardinality of a set.
The above diagram is a pictorial representation of a bijection, defined as a function that maps each point in set X to exact one point in set Y, in other words a one-to-one correspondence. It is clear that such a function, when applied to a set, will preserve its cardinality. For example, the function y=x-1 maps the set {2,3,4} to the set {1,2,3}, each of the sets having a cardinality of exactly 3. Since this function preserves the cardinality of all sets in the same matter, it is a bijection.
It is also possible to define an infinite set, or a set with an infinite number of elements. The simplest of these is the set of all natural numbers, namely {1,2,3,4...}. This set has infinite cardinality, and the cardinal number representing this set is the first so-called transfinite cardinal. It is denoted aleph-zero, or ℵ0.
Aleph-zero is not contained within the normal number system that we think of, but rather describes the size of the set containing all natural numbers. One could then wonder whether adding another element, for example 0, to the set would result in a cardinality of ℵ0+1. This intuitively seems reasonable, but it is not the case. It was earlier shown that the function y=x-1 is a bijection, so that, when applied to the above set X, will preserve its cardinality of aleph-zero. The result is as follows:
X={1,2,3,4...}
Y=X-1={0,1,2,3...}
Therefore, the set containing all whole numbers, which include both the natural numbers and 0, also has a cardinality of aleph-zero.
At first glance, the above result appears paradoxical. It seems that by subtracting all the terms by 1, an element is added to the beginning of the set, but taken off the end. This is certainly true for finite sets of natural numbers. For any finite natural number n, if
X={1,2,3...,n-1,n}, then
Y=X-1={0,1,2...,n-1}
However, as n increases without bound towards ∞, n-1 is ultimately indistinguishable from n, as ∞-1 is still ∞. In addition to this, when one considers any natural number in the infinite set {1,2,3,4...}, it is decreased by 1 when the function is applied, but there is always another number to take its place. When one takes these points into consideration, the result begins to make sense.
One can easily use the functions y=x-2, y=x-3... etc. to incorporate the elements -1, -2, etc. with the same logic as before, generalizing the above statement to include any finite number of negative integers.
This is only the beginning. Next, consider the function defined below.
If x is odd, then y=(x-1)/2
If x is even, then y=-x/2
The above is not a usual function that can be defined in simple operators of x. However, it is still a bijection when applied to the set of natural numbers, as each element of the set {1,2,3,4...} is transferred to a unique number in a second set. This set is {0,-1,1,-2,2-3,3...}. Remarkably, the output of this function covers all integers! Again, it is easy to see that for any integer one could choose, there is always a natural number that produces it with the above function. Therefore, the total set of integers still has the same cardinality, namely aleph-zero, as the natural numbers.
Nor does the fun stop there! Next consider the mapping of a set to a set of ordered pairs, namely assigning a set of two integers to each natural number. This can be done in the following way:
1 -> (0,0), 2 -> (1,0), 3 -> (1,1), 4 -> (0,1), 5 -> (-1,1), 6 -> (-1,0), 7 -> (-1,-1), 8 -> (0,-1), 9 -> (1,-1), 10 -> (2, -1)...
The exact pattern of these ordered pairs can take several different forms, the above being one of these. Initially, the above sequence seems to have no clear pattern, but it does have a clear geometric significance.
The above pattern lists all the integral ordered pairs (white circles) reached as one follows a rectangular counterclockwise spiral beginning at the origin (the black path). Clearly, by following this path for a sufficient distance, we will visit any ordered pair of integers (a,b) that we could choose!
Before exploring the implications of the above statement, we must make one more logical step. Consider an ordered triplet of the form (a,b,c), again with a, b, and c integers. One can use a similar system to the one above to set up a one-to-one correspondence between the natural numbers and these triplets. The first few terms are listed below:
1 -> (0,0,0), 2 -> (0,0,1), 3 -> (0,1,1), 4 -> (1,1,1), 5 -> (1,0,1), 6 -> (1,-1,1), 7 -> (0,-1,1), 8 -> (-1,-1,1), 9 -> (-1,0,1), 10 -> (-1,1,1)...
Just as before, this can be visualized as a rectangular spiral in the three dimensional coordinate system, where each point is given coordinates (x,y,z). In fact, this pattern, and its geometric interpretation, continue for any order n-tuplets, each consisting of elements (a1,a2,a3...an) and representing a spiral in the n-dimensional Cartesian system.
But what does this mean in terms of the number systems? Clearly, the set of all integral n-tuplets, for any finite n, has a cardinality of aleph-zero. From this, we draw the similar result that
Any infinite set or subset of a number system whose members can be represented by integral ordered n-tuplets, with n a natural number, has the same cardinality as the set of natural numbers, namely aleph-zero.
The implications of the above statement are explored in the next post.
Sources: http://en.wikipedia.org/wiki/Cardinal_number
Sunday, January 1, 2012
Juno
Juno is a NASA spacecraft whose mission is to orbit Jupiter and gain further insight to its composition and formation. It is named for the goddess Juno, wife of Jupiter in Roman mythology.
The spacecraft launched on August 5, 2011 to start its six year mission, culminating in a Jupiter arrival in 2016. The probe's trajectory included a flyby of Earth designed to conserve fuel. Unlike previous missions to the outer Solar System, Juno's energy will come only from solar panels, despite the relative dimness of the Sun at Jupiter's orbit.
In 2012, Juno executed several deep-space-maneuvers that prepared the probe for its flyby of Earth. Next, in October 2013, Juno completed its Earth flyby, assuming a trajectory directly toward Jupiter.
On July 4, 2016, the spacecraft executed an engine burn that inserted it into orbit around Jupiter. The probe assumed a highly elliptical orbit that took it past the north and south poles of Jupiter with every revolution.
The image above shows Juno's orbits around Jupiter over time, beginning with the orbital insertion on July 4.
This image, Juno's first acquired from orbit, shows the gas giant as well as three of the four Galilean moons, Io, Europa, and Ganymede (from left to right).
After its initial insertion burn, the Juno spacecraft spent over two months completing an elongated orbit that took it far away from the Solar System's largest planet. The first of 37 science flyby took place on August 27 and brought Juno over the north pole of Jupiter, capturing the first ever image of this polar region (see below).
The polar region is very different in appearance than the midlatitudes and equatorial region of Jupiter. The latter regions have characteristic colored bands of red, white, and orange, as well as prominent storm features. The poles are bluer, and lack these storm features. Juno's initial orbit was 53.4 days in duration. At its second closest approach to Jupiter on October 19, a maneuver was planned that would reduce the orbit to 14 days. However, the spacecraft entered safe mode just before the flyby when the onboard computer found conditions to be awry and neither data collection nor orbital maneuvering occurred on the 19th. Juno was later found to be functioning normally.
After two more successful flybys on December 11, 2016 and February 2, 2017, mission directors decided to not risk the reduction maneuver and maintain Juno in its 53-day orbit indefinitely. The main impediment to the function of the probe was the radiation belts near Jupiter's poles, which would gradually deteriorate Juno's functioning with every flyby. Since this radiation is only significant at closest approach, the longer orbit will not prevent the spacecraft from making the planned number of flybys. However, it did reduce their frequency by a factor of almost 4. Originally, 33 total orbits were planned in less than 1.5 years. The adapted budget plan covered only 12 orbits through July 2018, a span of two years.
Nevertheless, valuable data and images continued to pour in. The image below is a color-enhanced view of Jupiter's south pole, highlighting the massive swirling storms circling the pole.
In addition to images of the top of the atmosphere, Juno's instruments provided clues about deeper layers of the Jovian clouds. By the middle of 2017, enough Data had been collected from the Juno Microwave Radiometer, which detects thermal radiation from different depths in the atmosphere, to conclude that Jupiter's equatorial belts penetrated down to a great depth. In contrast, belts and storms at higher latitudes are relatively "shallow," with other structures appearing at increasing depth.
Nevertheless, the "weather layer" of Jupiter, which contains all the belts and cyclones, penetrates much further in depth than the analogous atmospheric layer on Earth. That is, the patterns of atmospheric movement (e.g. the spinning of Jupiter's great storms) persist down from the top portion of the atmosphere for a few thousand miles. In March 2018, four papers were published concerning Jupiter's atmospheric structure using Juno data. Among the results were the discovery of persistent circumpolar cyclones, as shown below.
The above two images are (false-color) computer generated composites of data from Juno's Jovian Infrared Auroral Mapper (JIRAM) instrument. The top image is the north polar region, showing a central cyclone surrounded by eight satellite cyclones. The south pole is similar, but has only five surrounding cyclones. Despite being in close proximity, these storms are much more persistent through time than those seen on Earth.
Another result published in this set of papers analyzed how Jupiter rotates below the weather layer. Precise gravitational measurements from the spacecraft indicate that a few thousand miles down into the atmosphere, the planet orbits approximately as a rigid body. That is, any deviations from steady rotation (such as the jet streams, belts, and storms) have a much, much smaller magnitude in this deeper layer.
In June 2018, NASA approved an extension of Juno's mission through 2021. Later in the year, Juno completed its global mapping of Jupiter with its 16th flyby of the planet. Each flyby had taken place at a different longitude (separated from the previous by 22.5°), allowing imaging of the entirety of the giant planet. By this time, 16 additional passes had been planned, offset from the first set to provide a composite total picture with finer resolution. During the first half of 2019, the Juno team had compiled enough measurements of Jupiter's magnetic field to determine that the field had changed measurably with time. Indeed, the measurements differed slightly but significantly from those of the Pioneer and Voyager spacecraft decades before. Nowhere other than the Earth had a changing magnetic field previously been detected. Experts expect that the variations stem from Jupiter's strong atmospheric winds, which move around material even in the deep layers containing molten metal. These inner layers drive the magnetic field just as on Earth.
A few months later, Juno executed a creative solution to a long-foreseen but very serious problem. As Juno explored different regions of Jupiter, it stood to reason that one flyby or another would bring the spacecraft into the giant planet's shadow. This posed a major problem for a solar-powered spacecraft, however, because only the batteries would prevent the temperature of various instruments from dropping too low. An unmodified orbital path on November 3, 2019, would have taken Juno into Jupiter's shadow for 12 hours, long enough to drain the batteries and possibly compromise the mission. Therefore, on September 30, the probe executed a 10.5 hour-long burn of its engines that allowed it to "jump the shadow" during the flyby a month afterward.
On its way to the December 26, 2019 flyby of Jupiter, Juno took images of Ganymede's north pole, shown below. The polar orbit of Juno let it see the poles more completely than previous space missions. At these poles were ice formations shaped by impacting charged particles directed toward the poles by Ganymede's magnetic field.
Sources: https://www.missionjuno.swri.edu/news/juno_spacecraft_in_orbit_around_mighty_jupiter, https://www.nasa.gov/feature/jpl/nasa-s-juno-spacecraft-sends-first-in-orbit-view, http://www.nytimes.com/2016/07/05/science/juno-enters-jupiters-orbit-capping-5-year-voyage.html?_r=0, https://www.nasa.gov/feature/jpl/jupiter-s-north-pole-unlike-anything-encountered-in-solar-system, https://www.nasaspaceflight.com/2016/09/juno-closest-approach-jupiter-readies-for-primary-science-mission/, https://www.nasa.gov/press-release/nasa-s-juno-mission-to-remain-in-current-orbit-at-jupiter, https://www.nasa.gov/press-release/a-whole-new-jupiter-first-science-results-from-nasa-s-juno-mission, https://www.nasa.gov/feature/jpl/nasa-juno-findings-jupiter-s-jet-streams-are-unearthly, https://www.nature.com/articles/nature25775, https://www.nasa.gov/feature/jpl/nasas-juno-mission-halfway-to-jupiter-science, https://www.missionjuno.swri.edu/news/Juno-Finds-Changes-in-Jupiters-Magnetic-Field, https://www.missionjuno.swri.edu/news/jun_prepares_to_jump_jupiters_shadow, https://www.nasa.gov/feature/jpl/nasa-juno-takes-first-images-of-jovian-moon-ganymedes-north-pole
The spacecraft launched on August 5, 2011 to start its six year mission, culminating in a Jupiter arrival in 2016. The probe's trajectory included a flyby of Earth designed to conserve fuel. Unlike previous missions to the outer Solar System, Juno's energy will come only from solar panels, despite the relative dimness of the Sun at Jupiter's orbit.
 |
| Juno's trajectory from launch in 2011 to arrival at Jupiter in 2016. |
In 2012, Juno executed several deep-space-maneuvers that prepared the probe for its flyby of Earth. Next, in October 2013, Juno completed its Earth flyby, assuming a trajectory directly toward Jupiter.
On July 4, 2016, the spacecraft executed an engine burn that inserted it into orbit around Jupiter. The probe assumed a highly elliptical orbit that took it past the north and south poles of Jupiter with every revolution.
The image above shows Juno's orbits around Jupiter over time, beginning with the orbital insertion on July 4.
This image, Juno's first acquired from orbit, shows the gas giant as well as three of the four Galilean moons, Io, Europa, and Ganymede (from left to right).
After its initial insertion burn, the Juno spacecraft spent over two months completing an elongated orbit that took it far away from the Solar System's largest planet. The first of 37 science flyby took place on August 27 and brought Juno over the north pole of Jupiter, capturing the first ever image of this polar region (see below).
The polar region is very different in appearance than the midlatitudes and equatorial region of Jupiter. The latter regions have characteristic colored bands of red, white, and orange, as well as prominent storm features. The poles are bluer, and lack these storm features. Juno's initial orbit was 53.4 days in duration. At its second closest approach to Jupiter on October 19, a maneuver was planned that would reduce the orbit to 14 days. However, the spacecraft entered safe mode just before the flyby when the onboard computer found conditions to be awry and neither data collection nor orbital maneuvering occurred on the 19th. Juno was later found to be functioning normally.
After two more successful flybys on December 11, 2016 and February 2, 2017, mission directors decided to not risk the reduction maneuver and maintain Juno in its 53-day orbit indefinitely. The main impediment to the function of the probe was the radiation belts near Jupiter's poles, which would gradually deteriorate Juno's functioning with every flyby. Since this radiation is only significant at closest approach, the longer orbit will not prevent the spacecraft from making the planned number of flybys. However, it did reduce their frequency by a factor of almost 4. Originally, 33 total orbits were planned in less than 1.5 years. The adapted budget plan covered only 12 orbits through July 2018, a span of two years.
Nevertheless, valuable data and images continued to pour in. The image below is a color-enhanced view of Jupiter's south pole, highlighting the massive swirling storms circling the pole.
In addition to images of the top of the atmosphere, Juno's instruments provided clues about deeper layers of the Jovian clouds. By the middle of 2017, enough Data had been collected from the Juno Microwave Radiometer, which detects thermal radiation from different depths in the atmosphere, to conclude that Jupiter's equatorial belts penetrated down to a great depth. In contrast, belts and storms at higher latitudes are relatively "shallow," with other structures appearing at increasing depth.
Nevertheless, the "weather layer" of Jupiter, which contains all the belts and cyclones, penetrates much further in depth than the analogous atmospheric layer on Earth. That is, the patterns of atmospheric movement (e.g. the spinning of Jupiter's great storms) persist down from the top portion of the atmosphere for a few thousand miles. In March 2018, four papers were published concerning Jupiter's atmospheric structure using Juno data. Among the results were the discovery of persistent circumpolar cyclones, as shown below.
The above two images are (false-color) computer generated composites of data from Juno's Jovian Infrared Auroral Mapper (JIRAM) instrument. The top image is the north polar region, showing a central cyclone surrounded by eight satellite cyclones. The south pole is similar, but has only five surrounding cyclones. Despite being in close proximity, these storms are much more persistent through time than those seen on Earth.
Another result published in this set of papers analyzed how Jupiter rotates below the weather layer. Precise gravitational measurements from the spacecraft indicate that a few thousand miles down into the atmosphere, the planet orbits approximately as a rigid body. That is, any deviations from steady rotation (such as the jet streams, belts, and storms) have a much, much smaller magnitude in this deeper layer.
In June 2018, NASA approved an extension of Juno's mission through 2021. Later in the year, Juno completed its global mapping of Jupiter with its 16th flyby of the planet. Each flyby had taken place at a different longitude (separated from the previous by 22.5°), allowing imaging of the entirety of the giant planet. By this time, 16 additional passes had been planned, offset from the first set to provide a composite total picture with finer resolution. During the first half of 2019, the Juno team had compiled enough measurements of Jupiter's magnetic field to determine that the field had changed measurably with time. Indeed, the measurements differed slightly but significantly from those of the Pioneer and Voyager spacecraft decades before. Nowhere other than the Earth had a changing magnetic field previously been detected. Experts expect that the variations stem from Jupiter's strong atmospheric winds, which move around material even in the deep layers containing molten metal. These inner layers drive the magnetic field just as on Earth.
A few months later, Juno executed a creative solution to a long-foreseen but very serious problem. As Juno explored different regions of Jupiter, it stood to reason that one flyby or another would bring the spacecraft into the giant planet's shadow. This posed a major problem for a solar-powered spacecraft, however, because only the batteries would prevent the temperature of various instruments from dropping too low. An unmodified orbital path on November 3, 2019, would have taken Juno into Jupiter's shadow for 12 hours, long enough to drain the batteries and possibly compromise the mission. Therefore, on September 30, the probe executed a 10.5 hour-long burn of its engines that allowed it to "jump the shadow" during the flyby a month afterward.
On its way to the December 26, 2019 flyby of Jupiter, Juno took images of Ganymede's north pole, shown below. The polar orbit of Juno let it see the poles more completely than previous space missions. At these poles were ice formations shaped by impacting charged particles directed toward the poles by Ganymede's magnetic field.
Sources: https://www.missionjuno.swri.edu/news/juno_spacecraft_in_orbit_around_mighty_jupiter, https://www.nasa.gov/feature/jpl/nasa-s-juno-spacecraft-sends-first-in-orbit-view, http://www.nytimes.com/2016/07/05/science/juno-enters-jupiters-orbit-capping-5-year-voyage.html?_r=0, https://www.nasa.gov/feature/jpl/jupiter-s-north-pole-unlike-anything-encountered-in-solar-system, https://www.nasaspaceflight.com/2016/09/juno-closest-approach-jupiter-readies-for-primary-science-mission/, https://www.nasa.gov/press-release/nasa-s-juno-mission-to-remain-in-current-orbit-at-jupiter, https://www.nasa.gov/press-release/a-whole-new-jupiter-first-science-results-from-nasa-s-juno-mission, https://www.nasa.gov/feature/jpl/nasa-juno-findings-jupiter-s-jet-streams-are-unearthly, https://www.nature.com/articles/nature25775, https://www.nasa.gov/feature/jpl/nasas-juno-mission-halfway-to-jupiter-science, https://www.missionjuno.swri.edu/news/Juno-Finds-Changes-in-Jupiters-Magnetic-Field, https://www.missionjuno.swri.edu/news/jun_prepares_to_jump_jupiters_shadow, https://www.nasa.gov/feature/jpl/nasa-juno-takes-first-images-of-jovian-moon-ganymedes-north-pole
Thursday, December 22, 2011
2011 Unnamed Tropical Storm
Storm Active: August 31-September 2
*This cyclone was classified as a tropical storm during the 2011 postseason analysis. It therefore received no name, despite being tabulated in the number of tropical depressions and tropical storms of the 2011 season.
On August 29, a circulation took shape in an area of convection north of Bermuda, some of which had been associated with Tropical Storm Jose a few days previously. The resulting trough organized over the next few days, increasing in shower activity. Late on August 31, a closed low formed on the southeastern edge of the shower activity, and the system became a tropical depression (although it was not then recognized as such). Convection increased markedly on September 1, and gale force winds were recorded, suggesting that the cyclone at this time became a tropical storm.
A banding feature to the southwest of the center formed the same day, and the cyclone strengthened overnight, reaching its peak intensity of 45 mph winds and a pressure of 1002 mb early on September 2. By this time, an approaching front had begun to push the system northeast, away from the U.S. east coast. The proximity of the front caused the unnamed storm to lose definition during the day of September 2, and the system became extratropical that evening. The remnants continued to move north-northeastward, and were fully absorbed on September 2. This event marked the first time since 2006 that a tropical storm was added in postseason analysis.
The unnamed tropical storm weakening on September 2.
Track of the unnamed tropical storm.
*This cyclone was classified as a tropical storm during the 2011 postseason analysis. It therefore received no name, despite being tabulated in the number of tropical depressions and tropical storms of the 2011 season.
On August 29, a circulation took shape in an area of convection north of Bermuda, some of which had been associated with Tropical Storm Jose a few days previously. The resulting trough organized over the next few days, increasing in shower activity. Late on August 31, a closed low formed on the southeastern edge of the shower activity, and the system became a tropical depression (although it was not then recognized as such). Convection increased markedly on September 1, and gale force winds were recorded, suggesting that the cyclone at this time became a tropical storm.
A banding feature to the southwest of the center formed the same day, and the cyclone strengthened overnight, reaching its peak intensity of 45 mph winds and a pressure of 1002 mb early on September 2. By this time, an approaching front had begun to push the system northeast, away from the U.S. east coast. The proximity of the front caused the unnamed storm to lose definition during the day of September 2, and the system became extratropical that evening. The remnants continued to move north-northeastward, and were fully absorbed on September 2. This event marked the first time since 2006 that a tropical storm was added in postseason analysis.
The unnamed tropical storm weakening on September 2.
Track of the unnamed tropical storm.
Thursday, December 15, 2011
2011 Season Summary
The 2011 Atlantic hurricane season was an above average season, with
20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status*
7 cyclones attaining hurricane status†
and 3 cyclones attaining major hurricane status
*In the NHC postseason analysis, an additional unnamed tropical storm was identified to have formed during the month of September. This means, that although 2011 only reached the letter "S" in tropical cyclone names, and 2010 reached "T", both seasons had the same number of tropical cyclones form.
†Nate was upgraded from a tropical storm to a hurricane during the postseason analysis.
At the beginning of the season, I predicted that there would be
20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status
10 cyclones attaining hurricane status
and 6 cyclones attaining major hurricane status
The tropical depression and tropical storm predictions happened to be exactly correct, although there was a lower number of hurricanes and major hurricanes than I predicted. As with the 2010 season, the 2011 season was tied for third in overall number of tropical storms with 19. This was caused by an ongoing La Nina event that actually intensified towards the latter part of the season. However, many of these storms were short-lived, and this reflects the abundance of favorable conditions for formation, but not for intensification. These conditions included high wind shear over much of the Caribbean for long periods of time, and also large pockets of dry air associated with anticyclones, which worked their way into many developing systems.
Some notable cyclones and facts about the season include:
Overall, the 2011 season was one of numerous, but weak, storms. The U.S. was affected much more than it had been in the previous two years with one hurricane and two tropical storm landfalls, but the damage associated with these systems was not severe.
20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status*
7 cyclones attaining hurricane status†
and 3 cyclones attaining major hurricane status
*In the NHC postseason analysis, an additional unnamed tropical storm was identified to have formed during the month of September. This means, that although 2011 only reached the letter "S" in tropical cyclone names, and 2010 reached "T", both seasons had the same number of tropical cyclones form.
†Nate was upgraded from a tropical storm to a hurricane during the postseason analysis.
At the beginning of the season, I predicted that there would be
20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status
10 cyclones attaining hurricane status
and 6 cyclones attaining major hurricane status
The tropical depression and tropical storm predictions happened to be exactly correct, although there was a lower number of hurricanes and major hurricanes than I predicted. As with the 2010 season, the 2011 season was tied for third in overall number of tropical storms with 19. This was caused by an ongoing La Nina event that actually intensified towards the latter part of the season. However, many of these storms were short-lived, and this reflects the abundance of favorable conditions for formation, but not for intensification. These conditions included high wind shear over much of the Caribbean for long periods of time, and also large pockets of dry air associated with anticyclones, which worked their way into many developing systems.
Some notable cyclones and facts about the season include:
- Hurricane Ophelia, the strongest storm of the season, attained Category 4 status at an unusually high latitude of 32.5° N
- 2011 was the first season in which none of the first eight tropical storms (Arlene through Harvey) became a hurricane
- Hurricane Irene, the first hurricane and major hurricane of the season, was also the first cyclone of hurricane strength to make landfall in the U.S. since Ike of 2008
- An unnamed tropical storm formed in early September, the first cyclone to be recognized only in the postseason analysis since 2006
- Nate was upgraded from a tropical storm to a hurricane in postseason analysis, the first such instance since 2007
- Hurricane Philippe was the longest lived storm in the Atlantic basin since 2008, but, despite its longevity, it affected no land.
Overall, the 2011 season was one of numerous, but weak, storms. The U.S. was affected much more than it had been in the previous two years with one hurricane and two tropical storm landfalls, but the damage associated with these systems was not severe.
Wednesday, November 9, 2011
Tropical Storm Sean (2011)
Storm Active: November 8-11
On November 4, a frontal boundary moved off of the U.S. east coast. A low pressure system along the front deepened as it moved off the coast of North Carolina later that day. The section of the front to the north of the low had most of the cloud cover associated with it, but the convection moved closer to the circulation of November 6, as the system drifted southeast. After temporarily losing definition, the low strengthened again on November 7. By this time, gale force winds occupied a region around the low, extending hundreds of miles in each direction.
Over the following day, the southern extension of the frontal boundary degenerated, devolving into a banding feature expanding clockwise from the low. Meanwhile, the remainder of the front had moved away to the east, and the leftover moisture became entrenched in the circulation of the cyclone. Early on November 8, convection had circumnavigated the center, and the system was upgraded to Subtropical Storm Sean.
The cyclone continued to increase in organization that afternoon, the eye contracting, and the surface circulation becoming better defined. The movement of the circulation into the lower levels of the atmosphere merited a reclassification of Sean into a tropical storm. Throughout the day, Sean remained nearly stationary, initially revolving around a broader cyclonic center, and later adopting a slow westward motion. Convection developed in earnest during the morning of November 9, and the system intensified into a strong tropical storm, also forming an eye feature.
By this time, Sean had entered the steering currents of the west Atlantic and began to accelerate northward. On November 10, the system curved to the northeast, also reaching its peak intensity of 65 mph winds and a pressure of 983 mb. Late that night, the windfield of Sean enveloped Bermuda, causing tropical storm force winds on the island, along with periods of heavy rain.
As it moved away from Bermuda on November 11 winds shear drastically increase and extratropcal transition began as Sean cam into close proximity with a front. By that evening, the center had elongated, and convective bands associated with the circulation ere stripped away. As a result, the system became extratropical that night. Sean caused minor damage and one fatality in Bermuda.
Tropical Storm Sean near peak intensity on November 10. The Outer Banks of North Carolina are visible on the upper left.
Erratic track of Sean through the Western Atlantic.
On November 4, a frontal boundary moved off of the U.S. east coast. A low pressure system along the front deepened as it moved off the coast of North Carolina later that day. The section of the front to the north of the low had most of the cloud cover associated with it, but the convection moved closer to the circulation of November 6, as the system drifted southeast. After temporarily losing definition, the low strengthened again on November 7. By this time, gale force winds occupied a region around the low, extending hundreds of miles in each direction.
Over the following day, the southern extension of the frontal boundary degenerated, devolving into a banding feature expanding clockwise from the low. Meanwhile, the remainder of the front had moved away to the east, and the leftover moisture became entrenched in the circulation of the cyclone. Early on November 8, convection had circumnavigated the center, and the system was upgraded to Subtropical Storm Sean.
The cyclone continued to increase in organization that afternoon, the eye contracting, and the surface circulation becoming better defined. The movement of the circulation into the lower levels of the atmosphere merited a reclassification of Sean into a tropical storm. Throughout the day, Sean remained nearly stationary, initially revolving around a broader cyclonic center, and later adopting a slow westward motion. Convection developed in earnest during the morning of November 9, and the system intensified into a strong tropical storm, also forming an eye feature.
By this time, Sean had entered the steering currents of the west Atlantic and began to accelerate northward. On November 10, the system curved to the northeast, also reaching its peak intensity of 65 mph winds and a pressure of 983 mb. Late that night, the windfield of Sean enveloped Bermuda, causing tropical storm force winds on the island, along with periods of heavy rain.
As it moved away from Bermuda on November 11 winds shear drastically increase and extratropcal transition began as Sean cam into close proximity with a front. By that evening, the center had elongated, and convective bands associated with the circulation ere stripped away. As a result, the system became extratropical that night. Sean caused minor damage and one fatality in Bermuda.
Tropical Storm Sean near peak intensity on November 10. The Outer Banks of North Carolina are visible on the upper left.
Erratic track of Sean through the Western Atlantic.
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