Infinity: The Aleph Sequence

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

In the previous seven posts, the world of countable ordinals has been explored in depth, allowing one to taste its incomprehensible vastness. It is truly remarkable how merely taking 2 to the power of ℵ₀ could produce a uncountable cardinal so "easily", but that no number of additions, multiplications, exponentiations, or functions on ω could yield an uncountable ordinal. It is through such an examination that one begins to appreciate how inconceivably numerous are the real numbers and the members of the continuum in comparison to the rational and even algebraic numbers. Yet it is possible to go still further.

The first uncountable ordinal is the set of all countable ordinals, which, not being a member of this set, confirms that it is not countable. It is denoted ω₁ (note the similarity in notation to ω₁^CK, the Church-Kleene ordinal defined previously). The cardinality of ω₁ is denoted ℵ₁, the second ℵ number. Additionally, ω₁ is what is called the initial ordinal of the cardinal number ℵ₁, as it is the smallest ordinal with this cardinality (analogously, the initial ordinal of ℵ₀ is ω).

Before going on, it is useful to describe another property that ordinals and cardinals may possess. Every ordinal (or cardinal) can be regular or singular. To determine whether any quantity is regular or singular, one must consider the set which has the given quantity as its supremum. For example, consider the ordinal ω². It is equal to

sup{ω,ω*2,ω*3,...,ω*n,...},

and the set in question, namely {ω,ω*2,ω*3,...,ω*n,...}, has order type ω (recall that ordinals themselves each represent an order type, and each order type represents a special class of sets that can be reached from one another with a one-to-one order preserving correspondence). Since the order type of this set, ω, is not equal to the original ordinal, ω², ω² is singular. Note that, though, the set above has the same cardinality as ω², it does not have the same order type. In contrast, ω is a regular ordinal. All the sets which have it as a supremum are infinite sets of natural numbers, and it is easily to see that they all have order type ω.

For cardinals, the definition is even simpler. One simply considers the sets of cardinals that have the cardinal as a supremum, and evaluate the cardinality of the sets. If these sets have the same cardinality of the number, it is regular. Otherwise, it is singular. For example, ℵ₀ is the first infinite cardinal, and is the supremum of any infinite sequence of finite cardinal numbers, each of which have a cardinality of ℵ₀. This cardinal is therefore regular. The order type or cardinality of the set of which the given quantity is a supremum is the quantity's cofinality. For a successor ordinal or cardinal, the cofinality is defined as 1, e.g. the ordinal 5 is the supremum of the set {4}, which has order type 1.

The first few regular ordinals are 0 (trivially), 1, ω, and ω₁. ω₁ can be seen as regular by the following argument. We know that countable sequences of countable ordinals have only countable ordinals as suprema. Therefore, the order type of a set {α,β,γ,...}, whose supremum is ω₁, is an uncountable ordinal. However, if this ordinal were greater than ω₁, some element of the set would also have to be greater than ω₁. Therefore, the cofinality of ω₁ is itself, and the first uncountable ordinal is regular. The cardinals corresponding to these four ordinals are also regular.

Recall the previous definition of the cardinal beth-one, which was that this cardinal was the cardinality of the real numbers, and that it was the cardinality of the power set of the set of all natural numbers, i.e. 2^ℵ₀. It has been proven that the statement "beth-one is equal to aleph-one" is independent of ZF theory. The proposition that this statement is true is known as the continuum hypothesis. In other words, the continuum hypothesis states that

2^ℵ₀ = ℵ₁

This seems reasonable, given that both cardinals in the above equality were encountered in the context of being the "first" uncountable cardinal. For the purposes of this post, the two will be assumed to be equal simply in order to avoid dealing with the beth-sequence separately. Also, the generalized continuum hypothesis, that each subsequent member of the beth-sequence is equal to its corresponding aleph cardinal, will be assumed for clarity. However, the focus on the development of cardinals will be based on that of ordinals, and little is lost by assuming that the aleph- and beth-sequences are synonymous.

The next cardinal number after ℵ₁ (assuming the continuum hypothesis), is ℵ₂, which has an initial ordinal ω₂, both of which are by definition regular for the same reasons as ω₁. Similarly, all ordinals between ω₁ and ω₂, of which there are uncountably many, are singular.

Similarly, one can define a series of regular cardinals ℵ₀, ℵ₁, ℵ₂, ℵ₃, and in general ℵ_n for natural n, and a analogous sequence of ordinals ω, ω₁, ω₂, ω₃,...,ω_n.

The cardinal sup{ℵ₀,ℵ₁,ℵ₂,...,ℵ_n,...} is denoted ℵ_ω. The initial ordinal of this cardinal is ω_ω. Since the cofinality ω_ω is ω (there is one member in {ω,ω₁,ω₂,...,ω_n,...} has one member for each natural n), this ordinal, and ℵ_ω, are singular.

ℵ_ω is also the first so-called inaccessible cardinal, meaning that its ordinal subscript is a limit ordinal. Similarly, larger cardinals, such as ℵ_ω*2, and ℵ_ω², having limit ordinals as indices, are inaccessible. Next, we "number" these inaccessible cardinals with ordinals. For example, ℵ_ω is identified with 1, ℵ_ω*2 with 2, and so on. Therefore, ℵ_ω² is labeled as the ω-th inaccessible cardinal. Similarly, every cardinal ℵ_ω*α is labeled α, as every limit ordinal is a multiple of ω.

A cardinal is considered 1-inaccessible if its initial ordinal is the same as its label on the list of inaccessible cardinals. This means that if the cardinal is of the form ℵ_ω*α, its initial ordinal must be α. The first cardinal for which this holds is the supremum of the set

{ℵ_ω,ℵ_ωω,ℵ_ωωω,...},
whose initial ordinal is ω_ωω....
Clearly, removing an "ω" from this infinite chain does not change the value, nor does multiplying by ω on the left. Therefore, this cardinal is 1-inaccessible. The first 1-inaccessible cardinal is the limit of the ℵ notation in a sense, as it requires an infinite number of "ω's" to write. Similarly, other 1-inaccessible cardinals exist, and can be numbered with ordinals in the same fashion. A cardinal whose initial ordinal is the same as its label on the list of 1-inaccessible cardinals is called 2-inaccessible. And so larger and larger cardinals are defined that are 3-inaccessible, 4-inaccessible. Clearly, for α-inaccessible cardinal κ, κ is also β-inaccessible for all β < α. Therefore, one can define a cardinal as ω-inaccessible if it is 1-, 2-, 3-,... inaccessible for all natural numbers.

Then, the notion of α-inaccessibility can be defined for any α, with the limit ordinal case being treated similarly to ω-for a limit ordinal α, a cardinal is α-inaccessible if it is also β-inaccessible for all β < α.

Next, a cardinal is hyper-inaccessible if it has an initial ordinal α, and is α-inaccessible. Such a cardinal is immensely large, as, for all cardinals yet considered, their initial ordinals were far higher than their degree of inaccessibility. For example, for the 1-accessible cardinal above, the degree of inaccessibility is only 1, while the initial ordinal α is so large that ω_α = α. By substituting "hyper-inaccessible" for "inaccessible" in the definition of 1-inaccessible cardinals, i.e. by numbering the hyper-inaccessible cardinals, one obtains corresponding definitions for 1-hyper-inaccessible, 2-hyper-inaccessible, and, in general α-hyper-inaccessible cardinals. Finally, there are hyper-hyper-inaccessible cardinals, the process can repeat, generating hyper-hyper-hyper-inaccessible cardinals, and so on.

A Mahlo cardinal is then defined as a cardinal that is inaccessible, hyper-inaccessible, and so on for any finite number of "hyper-'s". This is an example of a cardinal that cannot be proved to exist without the use of a large cardinal axiom. Without such an axiom, it is impossible to prove that any cardinal satisfies the needed property to be considered Mahlo.

As one explores higher cardinals, and their corresponding ordinals, each type of cardinal tends to become associated with the axiom with which its existence can be proved. To confirm that larger cardinals exist, one generally needs stronger and stronger axioms. However, if an axiom becomes too strong, it can sometimes create an inconsistent system (paired with other axioms), where a statement and its negative can both be proved true. If such an axiom was confirmed to prove an inconsistency, it seems reasonable that it would be discarded, and with it, its corresponding cardinals.

Therefore, in a way, there does exist an "upper limit" for cardinals and ordinals based on the strength of their accompanying axioms. Note that this does not mean there is a highest ordinal or cardinal, because this is impossible (e.g., for any ordinal α, its successor α' is greater than α and is also an ordinal). The situation is analogous to that of countable ordinals: there is no highest countable ordinal, yet the set of countable ordinals does have an upper limit, ω₁. In the case of the totality of cardinals and ordinals, there is probably even more "room to maneuver" below the upper limit, and plenty of cardinals not yet discovered.

The task of determining the hierarchy of large cardinals is ongoing and will not be discussed in detail here. However, the above description gives an idea of how such cardinals are defined. In conclusion, the realm of infinities, despite the insight provided by the use of cardinals and ordinals, is still not completely explored, and increasingly ingenious methods will be required to generate higher and higher quantities. The possibilities-as one would expect-are infinite!

Sources: Regular Cardinal, List of Cardinal Properties, etc., Wikipedia

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Author's Note (March 29, 2012): Of all my endeavors concerning this blog to date, the Infinity Series was perhaps the most ambitious, and definitely the most difficult. To attempt to convey even a glimpse of infinite sets and set theory is a challenging task. The above is far from an authoritative reference, and much is omitted where explanation would require the addition of large amounts of new material. Moreover, the mathematical arguments are less than rigorous, and the so-called "proofs" are often but sketches of their full versions-which require whole sets of axioms to complete.

Nevertheless, I am hopeful that this nuanced approach to a very narrow area of mathematics allows at least limited accessibility to those not acquainted with the nuts and bolts of set theory. For all my visitors, thank you for reading.

Professor Quibb