Infinity: Extended Collapsing Functions

This is the eleventh post of the Infinity Series. For the first, see here. For all posts, see the Infinity Series Portal.

In the previous post, an ordinal collapsing function was used to access high ordinals. However, all the outputs of the function discussed so far are alternatively expressible as Veblen functions with finite numbers of arguments. It was also seen that ψ(Ω^Ωα) is equal to the supremum of all ordinals that can be represented by an α + 1-argument Veblen function. This continues up to
ψ(Ω^Ωω), the first ordinal requiring infinite arguments to represent in Veblen notation. Being the supremum of countably many countable ordinals, it is countable, and is known as the small Veblen ordinal, though this ordinal is of course nothing near small.

In fact, it is small only when compared to even larger ordinals, such as the large Veblen ordinal, which is equal to ψ(Ω^ΩΩ). Alternatively, this ordinal represents the supremum of all others writable in Veblen notation of infinitely many, but only predicatively many, arguments. In short, since Veblen functions of high numbers of arguments are built off of those with lower numbers of arguments, only predicative ordinals, or those can be defined using lower ordinals alone, can serve as the number of arguments for such a function. Since Ω is clearly larger than any predicative ordinal, ψ(Ω^ΩΩ) is the supremum of the entire system of Veblen notation.

However, the ordinal collapsing function goes still further, defining ψ(Ω^ΩΩΩ), and in general ψ(Ω↑↑n) for finite n. Eventually one reaches the value ψ(ε_{Ω + 1}). The function is finally permanently stuck here, as ε_{Ω + 1} is not constructible from the original set S, and cannot be generated to plug to the ψ function. It is therefore never a member of the set C(α) no matter the value of α. ψ(ε_{Ω + 1}) is known as the Bachmann-Howard ordinal.

To go further using ordinal collapsing functions, we define a second function ψ₁, the value of ψ₁(α) being the smallest ordinal that cannot be finitely constructed using addition, multiplication, exponentiation, the original ψ function and ψ₁|_α, or the ψ₁ function restricted to α, these operations beginning from the set T₁ = {0,1,...,ω,...,Ω,Ω₂}, where Ω₂ is an ordinal higher than any of those that will be constructed with the ψ₁ function, and consequently, higher than Ω (again, it is not necessary for the definitions of Ω and Ω₂ to be in any way precise). Also, T₁ is assumed to contain all ordinals up to and including Ω, although we have no way yet of characterizing these ordinals, and so the definition is once again impredicative.

ψ₁(0) is clearly just ε_{Ω + 1}, as ψ₁|₀ is degenerate and has no values. But since all the ordinals less than Ω are in T₁, none of these can be equal to ψ₁(0), and since the ψ function only produces ordinals less than Ω, the first ordinal that is inaccessible is simply the limit of normal operations on Ω, namely ε_{Ω + 1}. Note that this is not the Bachmann-Howard ordinal, but a much larger indefinite quantity, the Bachmann-Howard ordinal being the ψ function at this value,
ψ(ε_{Ω + 1}). In fact, we cannot even be assured of the countability of ε_{Ω + 1}, due to the vagueness with which Ω was defined. Next,

ψ₁(1) = ε_{Ω + 2},

with the last term being the upper limit of iterated exponentiation on ε_{Ω + 1}. Now we return to the original collapsing function ψ, and modify its definition as follows:

ψ(α) is the first ordinal that cannot be constructed from the set S₁ = {0,1,ω,Ω,Ω₂} using the operations addition, multiplication, exponentiation, ψ|_α, and ψ₁.

Note no restriction is necessary on ψ₁ in this definition, as ψ₁ only produces values strictly greater than Ω, which is in turn greater than any ordinal produced by the ψ function. The outputs of ψ₁ therefore cannot affect which ordinal assumes the value of ψ(α), no matter its domain. This new definition produces values identical to those produced by the old for all ψ(α) such that α ≤ ε_{Ω + 1}, and using the same notation therefore produces no ambiguity. However, ψ(ε_{Ω + 1} + 1) is not equal to the Bachmann-Howard ordinal for this definition, since ε_{Ω + 1} is in C(ε_{Ω + 1} + 1). ψ(ε_{Ω + 1} + 1) is then equal to the next epsilon number after the Bachmann-Howard ordinal, namely

ε_{ψ(ψ1(0)) + 1} = ε_{ψ(εΩ + 1)) + 1}

One can go on through ψ(ψ₁(0) + 2), ψ(ψ₁(0) + Ω), and so on, on up to
ψ(ψ₁(1)) = ψ(ε_{Ω + 2}). Returning to ψ₁, ψ₁(2) = ε_{Ω + 3}, and in general
ψ₁(α) = ε_{Ω + 1 + α}, for all α up through Ω (with ψ₁(Ω) = ε_Ω*2) and beyond. Each of these can be plugged into the ψ function to yield yet another large countable ordinal, some of which are ψ(ψ₁(ψ₁(0))), ψ(ψ₁(ψ₁(ψ₁(0)))), and so on. Since each nested ψ₁ essentially iterates the ε function on Ω + 1, the limit of these expressions yields the first fixed point of such iteration, in other words the first α such that ψ₁(α) = α. This ordinal is ζ_{Ω + 1}, hence ψ₁(ζ_{Ω + 1}) = ζ_{Ω + 1}.

In a situation analogous to the original ψ function, ψ₁ is temporarily stuck at ζ_{Ω + 1}, as ζ_{Ω + 1} requires infinite steps to produce from T₁. ψ₁(α) is then fixed from ζ_{Ω + 1} on up to Ω₂, with ψ₁(Ω₂) still equal to ζ_{Ω + 1}, but ψ₁(Ω₂ + 1) is larger, as the domain of the restricted function ψ₁|_{Ω2 + 1} includes Ω₂, which is in T₁. ψ₁ is developed further in a similar way, a few notable values being

ψ₁(Ω₂*2) = ζ_{Ω + 2},
ψ₁(Ω₂²) = η_{Ω + 1}, and
ψ₁(ε_{Ω2 + 1}),

with the last being the highest value attainable by the ψ₁, as the function is constant afterward, in the same fashion as ψ is after ε_{Ω + 1}. Therefore,
ψ(ψ₁(ε_{Ω2 + 1})) is the highest ordinal accessible by the current definition of ψ, but is still much less than Ω.

Of course, there is no need to stop here. One can introduce a third function ψ₂, with ψ₂(α) the smallest ordinal not constructible from the set T₂ = {0,1,...,ω,...,Ω,...Ω₂,Ω₃}, in other words the set that contains all ordinals up to and including Ω₂, as well as an additional ordinal Ω₃, which is an arbitrary ordinal higher than any value that ψ₂ will attain.

ψ₂(0) is then equal to ψ(ψ₁(ε_{Ω2 + 1})). The definition of subsequent values of ψ₂ is as follows

ψ₂(α) is the smallest ordinal inaccessible from the operations of addition, multiplication, exponentiation, ψ, ψ₁, and ψ₂|_α from the set T₂.

One then alters the definitions of ψ₁, and finally, ψ, analogously in order to admit values such as ψ(ψ₁(ψ₂(Ω))), which is far greater than any ψ ordinal yet discussed, but of course considerably less than Ω. One can continue to generalize these functions, defining ψ₃, ψ₄, and so on, and each time altering all previous definitions accordingly. For natural numbers i and j, the function ψ_i adjusted for all ψ functions up to j, with j ≥ i, is written as ψ_ij. The general definition below is true for all functions ψ_ij (with the original ψ replaced by ψ₀ for clarity):

ψ_ij(α) is the smallest ordinal not constructible through a finite series of the operations addition, multiplication, exponentiation, ψ₀, ψ₁,..., ψ_i|_α, ψ_{i + 1},..., ψ_j on any elements of the set {0,1,...,ω,...,Ω,...Ω₂,...,Ω_i,Ω_{i + 1},...,Ω_j}, where each Ω_k is an arbitrary ordinal larger than any constructible with the function ψ_{k - 1}.

To understand this general definition, a few observations must be made. First, due to the definition of Ω_k, it is higher than any ordinal constructible with the previous ψ function. Therefore, the only restricted function in the definition is the one that is being defined, namely ψ_i, which produces ordinals strictly below Ω_i. Since the other ψ functions stay within their respective ranges also, they cannot (by themselves) produce a value in this range, and cannot directly affect ψ_i(α). Also, the purpose of the value j is simply to identify how "upgraded" each ψ function is, e.g., the difference between the initial definition of ψ (ψ₀) versus its "upgraded" definition incorporating ψ₁. Moreover, the set from which the ordinals are being constructed includes all ordinals up through Ω_i, but afterwards only the Ω ordinals between i and j.

Finally, the purpose of this multitude of functions is merely to plug them back in to ψ in order to generate further values, all of which are still far less than Ω. The limit of these functions is still less than Ω, and is sometimes denoted ψ_ω(0). By the use of fixed points, many more ordinals can be defined in this matter, but the processes involved are identical to those discussed. However, there are still higher countable ordinals, some of which are even beyond the principle of recursion (see the next post).

Sources: Large Countable Ordinals- Wikipedia, http://www.ams.org/journals/tran/1908-009-03/S0002-9947-1908-1500814-9/S0002-9947-1908-1500814-9.pdf

Infinity: Impredicative Ordinals

This is the tenth post of the Infinity Series. For the first, see here. For all posts, see the Infinity Series Portal.

At the conclusion of the previous post, Γ₀, the first impredicative ordinal, and the first defined with a "second-order" definition, was introduced. The remainder of the Γ-sequence was also defined. To find higher countable ordinals, one precedes in a similar manner to previously, by finding fixed points.

For example, the smallest fixed point of α → Γ_α is also the supremum of the set {Γ₀,Γ_Γ0,Γ_ΓΓ0,...}, and is known as the Ackermann ordinal, named after Wilhelm Ackermann, known for his contributions to computation theory.

Again, it becomes useful to have a standard notation for these ordinals, and a generalized form of the Veblen hierarchy is quite useful. To extend the Veblen notation, first we replace what was previously denoted as φ_β(α) with φ(α,β), namely a function of two arguments, the first being what was before the argument, and the second that which was the subscript. To extend the Veblen functions beyond what was already considered, we add a third argument γ, and consider the ordinals defined by expressions of the form φ(α,β,γ). These expressions are evaluated in the following way:

First, φ(α,β,0) = φ(α,β). Then, φ(0,0,1) is defined as the first ordinal that is inaccessible through the expressions φ(α,β), constructing from 0 upward. Since this definition is identical to the one for the Feferman-Schutte orindal, φ(0,0,1) = Γ₀. To become more familiar with the notation, consider these further examples:

φ(0,1,1) = φ_{Γ0 + 1}(0), or the first fixed point of the first Veblen function after φ_Γ0, φ(0,0,2) is equal to Γ₁, or the second fixed point of φ_α(0) → α. Every increase here in the second argument of the generalized Veblen function represents listing fixed points. Each increase in the third represents another Γ value. For example, φ(0,0,2) = sup{φ(0,0,1),φ(0,φ(0,0,1),1),φ(0,φ(0,φ(0,0,1),1),1),...}, with the second argument β assuming the value of the previous member in each element of the set, until Γ₁ is obtained. It is easy to see this is consistent with the previously cited value of Γ₁. All other φ(0,0,γ) can be defined this way, and, continuing from the discussion of Γ₀, which was defined in a second-order fashion, these ordinals have second-order definitions, and are all impredicative.

The smallest ordinal inaccessible from the standard operations on all the ordinals φ(0,0,1), φ(0,0,2), φ(0,0,3), etc. of finite third argument γ, yields a new γ value: ω, the ordinal being written φ(0,0,ω). Of course other values for α and β are also possible, such as φ(1,5,ω) or φ(ε₀,Γ₃,ω). But of more concern is the third argument, which can increase through more countable values until reaching nested Veblen functions such as φ(0,0,φ(0,0,1)), defining a sequence in which the γ assumes the value of the previous member of the sequence. Having gone as far as possible with three arguments, a fourth is introduced in the following way:

sup(φ(0,0,1),φ(0,0,φ(0,0,1)),φ(0,0,(φ(0,0,φ(0,0,1))),...} = φ(0,0,0,1)

Since φ(0,0,1) = Γ₀, the above equation is exactly the same as sup{Γ₀,Γ_Γ0,Γ_ΓΓ0,...} = φ(0,0,0,1). Additionally, φ(0,0,0,1) is the first fixed point of Γ_α → α, and, as said above, is also known as the Ackermann ordinal. This ordinal has a "third-order" definition, and is also impredicative by most definitions.

One can go on to define φ(α,β,γ,...,δ) for any finite number of arguments, all of which are limits similar to the example above, and although inaccessible from previous operations, are countable for all countable α,β,γ,...,δ. It becomes inconvenient to deal with so many arguments however, so there is an alternate method to create large countable ordinals with a function of only one argument.

As stated above, it is possible to go still further, by the use of a new kind of function, called an ordinal collapsing function. These functions again use the idea of inaccessibility in their definition, but there is a crucial difference between these and Veblen functions: ordinal collapsing functions assume the existence of an ordinal Ω that is beyond any ordinals that can be constructed with ordinal collapsing functions. This ordinal does not have to be defined in any precise way, but simply is assumed to exist as an upper limit to any ordinal constructed. However, since ordinals generated by ordinal collapsing functions make use of Ω, this definition is impredicative, as it assumes the existence of a larger ordinal to define a smaller.

To construct this function, first consider the set of ordinals S = {0,1,ω,Ω}. The set C(0) is then defined as containing all ordinals that can be constructed through a finite process from 0,1,ω, and Ω using addition, multiplication and exponentiation, including 1, 2, 3, ω, ω^ω, and ω^ωω. C(0) also includes Ω, Ω + 1, Ω^Ω, and others of this type, but these are not as important for the time being.

Now, we denote the smallest ordinal that is not in C(0) as ψ(0) (with the symbol psi, rather than phi for Veblen functions). ψ(0) is then equal to ε₀, the first ordinal that requires infinite steps from ω to construct. Next, C(1) is the set of ordinals constructible from S with addition, multiplication, and exponentiation, and the function ψ|₁, or the ψ function restricted to 1. Such a function can only take as an argument values that are members of 1, namely, only 0 (all ordinals are sets of all previous ordinals). Hence the ordinal ψ(0) = ε₀ is a member of C(1), as well as sequences of the normal three operations on it, such as ε₀ + 1, ε₀^ε₀, and so on.

The smallest ordinal not in C(1) is written ψ(1), and is equal to ε₁, as no finite sequence of operations from ε₀ yields this value. Similarly, C(2) is the set of ordinals accessible from members of S through the normal operations plus ψ|₂. Since 1 is a member of 2, ψ(1) = ε₁ is a member of C(2), as is ε₀ and any operations on these two ordinals with the other members of S. Preceding recursively, C(α) has as a member all ordinals defined through the three normal operations, and ψ|_α, which has a domain of all ordinals less than α.
ψ(α) is defined as before, and is seen to equal ε_α for all finite α.

But what about when α = ω? C(ω) includes all values taken by ψ|_ω, namely ε₀, ε₁, ε₂,..., ε_α, and so on for all finite α, as well as operations upon them. The corresponding ψ(ω) is the clearly ε_ω as this transcends all previous epsilon values and operations on them. Thus the equivalence of ψ(α) and ε_α continues for some infinite values of α. In fact, this equality holds true as long as α is less then or equal to ε_α.

However, in the case of α = ζ₀ (for which ε_α = α), C(ζ₀) includes all of the epsilon sequence below ζ₀, but does not contain the ordinal itself, as even applying the function ψ|_ζ0 to 0 any finite number of times (producing the sequence ε₀, ε_ε0,...) cannot yield ζ₀, and ψ(ζ₀) = ζ₀. α = ζ₀ + 1 presents an interesting case. One would think that the pattern would continue, generating ε_{ζ0 + 1} as the value for ψ(ζ₀ + 1). However, for this to happen, ζ₀ would have to be a member of C(ζ₀), since if this were not the case, ε_{ζ0 + 1} would not be the smallest ordinal not in C(ζ₀). But, beginning with the set S, it is impossible to construct ζ₀ with a finite number of operations. Even though one of the operations available for C(ζ₀) is ψ|_{ζ0 + 1}, which has ζ₀ in its domain and would produce ζ₀, no number of steps from S can produce this value in the first place to plug it into the function! There is therefore no way to obtain ζ₀, and
ψ(ζ₀ + 1) = ζ₀.

By the same logic, ψ(ζ₀ + 2), ψ(ζ₀ + ω), and others are still equal to ζ₀. It may seem that the function is "stuck" at ζ₀, and can produce no higher value. This is where Ω comes in. Consider the set C(Ω). This set is no different from previous ordinals in that it ψ(Ω) = ζ₀. The difference appears when considering C(Ω + 1), as ordinals in this set can be constructed from S using ψ|_{Ω + 1} along with the other three operations. Since Ω is in S and in the domain of ψ|_{Ω + 1}, ψ(Ω), that is, ζ₀, is in C(Ω + 1). If not for the addition of Ω to S, this function would cease to increase, and this is the importance of Ω. It also explains the name "ordinal collapsing function", as it takes large ordinals such as Ω, and using a function, "collapses" them into smaller ordinals, in this case ζ₀.

It has been established that ψ(Ω + 1) is larger than ζ₀, but what is it? C(Ω + 1) also includes the usual operations on ζ₀, including iterated exponents such as ζ₀^ζ₀ζ₀. Clearly the smallest ordinal that cannot be accessed in this way is ε_{ζ0 + 1}, and this is equal to ψ(Ω + 1). The correlation between the ε and ψ functions continues from there, with ψ(Ω + α) = ε_{ζ0 + α} for all finite and some infinite values of α, including cases greater than ζ₀. For example, ψ(Ω + ζ₀*2) =
ε_{ζ0 + ζ0*2} = ε_ζ0*3.

The above trend occurs up to ψ(Ω + ζ₁), but the function ψ|_{Ω + ζ1} cannot equal ζ₁ for any ordinal in its domain, nor can iterated exponentiation generate this value. Thus, ψ(Ω + ζ₁) = ζ₁. The next problem occurs when one reaches ψ(Ω + ζ₁), as the set C(Ω + ζ₁) has the operation ψ|_{Ω + ζ1 + 1}, which gives ζ₁, but only for the input Ω + ζ₁, and this input cannot be constructed with other operations to plug into the function. Thus, ψ(Ω + ζ₁) = ζ₁. Hence the function is stuck once again, with ψ(Ω + α) being equal to ζ₁ for α greater than or equal to ζ₁, but only up to a certain point.

Again, Ω bails us out, or, more precisely, Ω*2. Though ψ(Ω*2) still is ζ₁, Ω*2 can be constructed easily from S, and the set C(Ω*2 + 1) has available the operation ψ|_{Ω*2 + 1}, which has Ω*2 in its domain, and can produce ζ₁. Through similar logic as above, ψ(Ω*2 + 1) = ε_{ζ1 + 1}. The general idea continues, with each ψ(Ω*(1 + α) + 1) transcending another ζ ordinal, and yielding ε_{ζα + 1} (if this formula is unclear, try substituting 0 and 1 for α, which give known values, and serve as examples).

As it turns out, this repetition of multiples of Ω "unsticking" the function continues all the way up to ψ(Ω*η₀), with η₀ being the first fixed point of
α → ζ_α. ψ(Ω*η₀) is equal to η₀, as finite iterations of α → ζ_α from 0 give ordinals strictly less then this value. However, ψ(Ω*η₀ + 1) is also η₀, and the function is stuck again, this time all the way until ψ(Ω²), with ψ(Ω² + 1) equal to ε_{η0 + 1}, the first epsilon ordinal greater than η₀. In fact, further powers of Ω have ψ values corresponding roughly to Veblen functions of two arguments, up to
ψ(Ω^Γ₀) = ψ(Ω^Ω) = Γ₀, with the function stuck at Γ₀ for all ψ(Ω^α) for Γ₀ ≤ α ≤ Ω. ψ(Ω^Ω) is then the same as φ(0,0,1) = Γ₀. Multiplying by Ω corresponds to raising the second argument of the Veblen function, and multiplying the exponent of Ω by Ω raises the third argument. For example, ψ(Ω^{Ω + 1}) is
φ(0,0,2), or Γ₁. More values include:

ψ(Ω^Ω2) = φ(0,0,0,1), or the Ackermann ordinal, ψ(Ω^Ω3) = φ(0,0,0,0,1), and so on, with each ψ(Ω^Ωα) equal to the smallest ordinal requiring α + 2 arguments to represent. Alternatively, each of these is the supremum of all ordinals requiring at most α + 1 arguments to represent. However, the ordinal collapsing function has only yielded ordinals expressible as Veblen functions of finite arguments. Even higher countable ordinals accessible through this function are discussed in the next post.

Sources: http://www.ams.org/journals/tran/1908-009-03/S0002-9947-1908-1500814-9/S0002-9947-1908-1500814-9.pdf, Ordinal Collapsing Function - Wikipedia

Infinity: Larger Countable Ordinals

This is the ninth post of the Infinity Series. The first is found here. For all posts, see the Infinity Series Portal.

Over the course of the previous post, a plethora of infinite ordinals were introduced, all of which were, somewhat surprisingly, countable. We concluded with the epsilon sequence, a series of ordinals α for which ω^α = α, where ω is the set of all finite ordinals, and therefore the smallest infinite ordinal.

It was also discussed that any epsilon sequence member of countable index was itself countable. Therefore, ε_ε0, ε_ε1, ε_εω, and so on, are countable.

Since ε_ε0 is countable, the epsilon ordinal with this index, namely ε_εε0, is also countable, and one can continue this recursive process indefinitely. By doing this, one can construct the set {ε₀,ε_ε0,ε_εε0,...}, with each member of the set having a finite number of "ε's". Since there is one and only one member of this set for every positive natural number, it is countable, and again has a countable supremum. This ordinal is sometimes called ζ₀ (zeta-naught), and is also defined as the smallest fixed point of the function on ordinals which carries each ordinal α to ε_α. Hence ε_ζ0 = ζ₀.

Ordinals accessed in this way, such as ε₀ and ζ₀ are values of what are called the Veblen functions, which are functions that generate fixed points of given other functions. In this case, ε₀ is a value of the Veblen function on α → ω^α and ζ₀ a value of the Veblen function on α → ε_α.

Of course, ζ₀ is only the smallest fixed point of the second Veblen function, and is certainly not the end of countable ordinals. ζ₀ + 1, ζ₀ + 2, ζ₀*2, ζ₀^ω and other normal operations on ζ₀ can be defined, eventually leading one to the ordinal ε_{ζ0 + 1}, which, by the definition of the epsilon sequence is the limit of the sequence ε_ζ0, ε_ζ0^ε_ζ0, and so on. However, ε_ζ0 is simply equal to ζ₀, so

ε_{ζ0 + 1} = sup{ζ₀,ζ₀^ζ₀,ζ₀^ζ₀ζ₀,...,ζ₀↑↑n,...}.

Hence this latest ordinal is countable, as are ε_{ζ0 + 2}, ε_ζ0*ε0, and others, eventually leading up to ε_{εζ0 + 1}. Being countable, the epsilon ordinal of this index is as well, in addition to any subsequent iterations.

Thus, one can define the countable ordinal sup{ζ₀ + 1,ε_{ζ0 + 1},ε_{εζ0 + 1},ε_{εεζ0 + 1},...}. In the limit, this is clearly another fixed point of the second Veblen function, but it is greater than ζ₀, as it incorporates ζ₀ + 1 into its definition. It is then the second fixed point of α → ε_α, and is denoted ζ₁. Continuing in this manner, ζ₂ = sup{ζ₁ + 1,ε_{ζ1 + 1},ε_{εζ1 + 1},...}, where ζ₂ is again a fixed point of the above function, i.e. ε_ζ2 = ζ₂. Repeating this process, one obtains ζ_α for any finite α, and, in the limit of these, ζ_ω. For the same reasons as the epsilons, any zeta ordinal of countable index is countable. We then have the countable ordinals ζ_ζ0, ζ_ζζ0, etc., culminating in a new ordinal, namely the smallest fixed point of the function α → ζ_α, and the smallest value of the Veblen function on this mapping.

Although this ordinal could be written η₀ (eta-naught), it becomes useful to have a more standard notation for Veblen functions. To do this, first denote the function α → ω^α as φ₀. Then, for example, φ₀(1) = ω¹ = ω, and φ₀(3) = ω³. Moving on, the function φ₁ is defined as the Veblen function on φ₀, and therefore "lists" its fixed points. φ₁(0) then means "the first fixed point of α → ω^α", which we know to be equal to ε₀. φ₁(1) is the second fixed point of φ₀, which is ε₁. It is clear from the definition of the epsilon sequence that φ₁(α) = ε_α. To find subsequent Veblen functions, one uses the recursive scheme φ_β' (The Veblen function indexed by the successor of β) = "the function that lists the fixed points of φ_β". This series of functions is known as the Veblen hierarchy. Therefore, φ₂ lists the fixed points of φ₁, and is equivalent to the function α → ζ_α. Similarly, φ₃(0) is the first fixed point of φ₂, namely what we briefly referred to as η₀.

Any function of the Veblen hierarchy has a domain of the ordinal numbers, with countable arguments generating countable outputs. Using the recursive formula above, the functions φ₄, φ₅, etc., can be constructed. Another important fact about Veblen functions is that each value taken by one of these functions is taken by all previous functions. In formula form, where α, β, γ, and δ and ordinals,

For all ordinals φ_β(α), there exists an ordinal γ for all δ < β such that φ_β(α) = φ_δ(γ)

In other words, each value taken by a function in the Veblen hierarchy is also a fixed point of all previous ones. For example, ζ₀ is also a fixed point of α → ω^α, as well as of α → ε_α. Using this fact, φ_β(α) can be defined in the case that β is a limit ordinal:

If β is a limit ordinal, then φ_β lists the common fixed points of all the functions φ_δ for all δ < β. For example, φ_ω(0) is the smallest ordinal that is a fixed point of all the functions φ₀, φ₁, φ₂, etc. One can further define the ordinals φ_ε0(0), φ_ζ0(0), and even

φ_φω(0)(0), φ_φφω(0)(0)(0), and so on.

All ordinals accessible through the operations of addition, multiplication, exponentiation and the Veblen hierarchy have now been described. Next, we define the ordinal that is the limit of the above ordinals, i.e. the smallest ordinal such that φ_α(0) = α. It is also described as the first fixed point of
α → φ_α(0). This ordinal transcends the entire Veblen hierarchy, and it is sometimes called the Feferman-Schutte ordinal. It is written Γ₀.

Γ₀ has many interesting properties. First and foremost, it is countable, being constructed using Veblen ordinals of countable argument. Also, it is considered by many to be what is called "impredicative", essentially meaning here that it cannot be defined without reference to the totality of the system of ordinals. This is because, unlike all previous ordinals, it depends on a "second-order" definition, namely it is dependent on the Veblen hierarchy, each member of which is a function. In this way, it is constructed through a sort of "meta"-function, i.e. a function on functions. Since functions such as those of the Veblen hierarchy have domains encompassing the system of ordinal numbers, Γ₀ makes use of this entire system in its construction, and references all ordinal numbers, including itself. The fundamental point is that the definition that led to Γ₀ was of a different nature than all those previously, and is the first of a different class of ordinals, the impredicative ordinals.

Also, by definition of its construction, it is a fixed point of all Veblen functions φ_β, where β < Γ₀. Hence ω^Γ₀, ε_Γ0, ζ_Γ0, etc. are all equal to Γ₀. We use this fact to confirm that

sup(Γ₀ + 1,ω^{Γ₀ + 1},ω^{ωΓ₀ + 1},..} = ε_{Γ0 + 1} > Γ₀.

Similarly, sup(Γ₀ + 1,ε_{Γ0 + 1},ε_{εΓ0 + 1},...} = ζ_{Γ0 + 1} > Γ₀.

Further, φ_β(Γ₀ + 1) can be defined for higher ordinals, eventually leading to φ_{φΓ0 + 1(0)}(0). The limit of

φ_{Γ0 + 1}(0), φ_{φΓ0 + 1(0)}(0),...

is then the second fixed point of φ_α = α, and, being greater than Γ₀, it is denoted Γ₁. In the same way, Γ₂, Γ_ω, Γ_Γ0 can be defined. Being limits involving Veblen functions, these ordinals are also countable. However, these still does not constitute all of the countable ordinals. The trek forward is continued in the next post.

Sources: http://folk.uio.no/herman/incompleteness.pdf, Large Countable Ordinals-Wikipedia

Infinity: Countable Ordinals

This is the eighth post of the Infinity Series. For the first post, see here. For all posts, see the Infinity Series Portal.

In the previous posts (here and here), ordinal numbers and basic operations concerning them were introduced. Now, using exponentiation, the ordinal ω² can be defined. By the definition of exponentiation,

ω² = ω*ω,

which is greater than any ordinal ω*n, for all natural numbers n, and is therefore greater than ω. ω² as a set is written {0,1,2,...,ω,ω + 1,...,ω*2,ω*2 + 1,...,ω*3,...,ω*n,...}.

In contrast, consider 2^ω, which is the limit of 2ⁿ over natural numbers. Clearly, this is just the limit of a sequence of increasing natural numbers, namely ω. Therefore, 2^ω = ω. In the same way, one can define further ordinals of the form ωⁿ, and confirm that n^ω = ω for all natural numbers n.

To prove that ordinals such as ωⁿ are countable, one must use the following theorem:

The union of countably many countable sets is countable. (1)

Besides being a mouthful, this theorem is invaluable in confirming the countability of sets. To see how the theorem works, consider a countable set of countable sets {A,B,C...} (each of A, B, C, etc. is a countable set). Since this set is countable, the sets in this set can be labeled with natural numbers. Each set can then be identified with, and written, as a number, (for clarity, the numbers will be in boldface), with A = 1, B = 2, C = 3, and so on. Since each of these sets individually is countable, the elements of each one can be labeled 1, 2, 3 and so on. For example, the element labeled "1" in set A will be denoted 1₁. We now construct the following function for labeling the elements of all of the above sets (the first ten values, corresponding to the natural numbers 1-10, are shown)

1→ 1₁, 2→ 1₂, 3→ 2₁, 4→ 1₃, 5→ 2₂, 6→ 3₁, 7→ 1₄, 8→ 2₃, 9→ 3₂, 10→ 4₁,...

It is easily calculated that the general member m of any set n can be found by the following formula:

((m + n)(m + n - 1))/2 - n + 1→ n_m

This formula may seem rather complicated, but the important fact is that for any element of any set in {A,B,C...}, there is exactly one natural number that serves as its "label", and the above sequence is a one-to-one correspondence.

We first observe that ω² is countable, as it is a union of all sets of the form {ω*n + 0, ω*n + 1,...} for natural n. Each of these is countable, as every set of the above form can be put into a one-to-one correspondence with the natural numbers simply by dropping the ω term from each member, and the set of all these sets is also countable, as there is one for each natural number n. (1) is therefore applicable, and ω² is countable.

Similarly, one can define a set of sets of the form {(ω²)*n,(ω²)*n + 1,...,(ω²)*n + ω,(ω²)*n + ω + 1,...,(ω²)*n + ω*2,...,(ω²)*n + ω*3,...,(ω²)*n + ω*m,...}. If n = 0, then the set above reduces to ω². For all other values of n, the set can be put into a one-to-one correspondence with ω² by dropping the first term. The union of all these sets, with the set of sets being, of course, countable, is ω³.

The above outlines a recursive scheme that confirms that all ωⁿ are countable. The countability of each power of ω allows one to apply (1) to the next power. Therefore, if one repeats this process indefinitely over all natural numbers, the limit is the ordinal ω^ω, which is again countable, being a limit of countably many countable sets. The property of such a limit being countable is very important to ordinal numbers and will be used often. It is a consequence of the theorem proved above, and will be referred to by (2).

This fact seems at odds with what was previously discussed about cardinal numbers, with 2^ℵ₀ being equal to the cardinality of the continuum, which describes a property of uncountable sets. It seems much "easier" in a sense to access uncountable cardinals than it is to describe uncountable ordinals, and it will soon be described just how difficult the latter is.

Continuing from ω^ω, one can use a similar recursive scheme to show that ω^ω2, ω^ω3, and so on, are countable. Remarkably, under the same limiting process, combined with recursion, ω^ωω, ω^ωωω, and any larger finite power towers of ω are countable.

At this point, no larger ordinals can be described through the operations of addition, multiplication, and exponentiation directly in terms of ω. We therefore define the ordinal ε₀ (epsilon-nought) as the limit of the sequence ω, ω^ω, ω^ωω, ω^ωωω,..., or, more formally, as the smallest ordinal α such that ω^α = α. In other words, this ordinal is so large that the two quantities ω^ε₀ and ε₀ are indistinguishable. Mathematically, ε₀ is also the smallest fixed point of the map α → ω^α, i.e. is left unchanged when it is applied.

Remarkably, ε₀, despite being extremely large, is still countable! It is the limit of the set {ω, ω^ω, ω^ωω, ω^ωωω,...,ω↑↑n,...}, where ↑↑ represents the hyper4 operator, or iterated exponentiation, which in this case a shorthand for writing "n ω's in the power tower". Since, in this set, n takes only natural number values, the set itself is countable, and ε₀ is the limit of countably many countable ordinals, and, by (2), is itself countable.

Of course, ordinals do not end here. One can go on to define ε₀ + 1, ε₀*2, ε₀^ω and others. For the same reasons as the corresponding expressions involving ω, these are all countable. One can go to on to define the limit of the set {ε₀,ε₀^ε₀,ε₀^ε₀ε₀,...,ε₀↑↑n,...} as a new ordinal, ε₁. Reviewing the process through which this ordinal was created, one can see that it is a limit of a set of limit ordinals, each of which was itself a limit of a sequence of ordinals! Despite being so far removed from our original ω, it is again countable, by the same logic used for ε₀. Also, this is the second fixed point of the map mentioned above, as all ordinals between ε₀ and ε₁ are not equal to themselves under
α→ ω^α, e.g. ω^{ε₀ + 1} = ω^ε₀*ω = ε₀*ω ≠ ε₀ + 1.

Continuing even further, one can define ε_α for any natural number α. The formula below shows how this is done, with the symbol "sup{}", representing the supremum, or least ordinal greater than or equal to every member, of a set. Essentially, the notation "supremum of a set" is interchangeable with "limit of a sequence", but the former is easier to write. The formula for ε_α is recursive, using succession to define each member of the epsilon sequence.

ε_α' = sup{ε_α,ε_α^ε_α,ε_α^ε_αε_α,...,ε_α↑↑n,...}

Since each set in the above equation is countable, and each ordinal involved is countable, the members of the epsilon sequence for finite α are countable. Of course, one can then construct a set of all epsilon ordinals of natural number index α, i.e. {ε₀,ε₁,ε₂,...,ε_α,...}. This set is countable, as well as all of its members, and it therefore has a countable supremum, which can be defined as ε_ω. Subsequently,
ε_{ω + 1}, ε_{ω + 2},...,ε_ω*2, and others can be defined. Each of these ordinals, if the subscript of ε is limit ordinal, is simply the supremum of all previous members of the epsilon sequence. Otherwise, the subscript is the successor to some other ordinal, and the formula that was used for ε ordinals of finite indices can be applied.

An important property of the epsilon sequence is that any ordinal in the sequence with a countable index is also countable. This is because, for such an ordinal, the set of all previous members of the epsilon sequence is countable. The reason for this is that they can be easily put into a one-to-one correspondence with a countable set, simply by dropping the "ε", and considering only their indices. For example, ε_ω*2 is the supremum of the set
A = {ε₀,ε₁,ε₂,...,ε_ω,ε_{ω + 1},...}. A bijection can be constructed from this set to ω*2, namely the one assigning ε₀→0, ε₁→1, etc., confirming the countability of A and the applicability of (2).

Among the epsilon ordinals of countable index are ε_ω^ω, ε_ω^ωω, and even ε_ε0. But where can one go from here? This is still just the beginning of infinite ordinals. More are explored in the next post.

Sources: Axiomatic Set Theory by Patrick Suppes, http://en.wikipedia.org/wiki/Epsilon_nought, Large Numbers at MROB