Inaccessible cardinal

There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible.

Weakly inaccessible

Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the continuum hypothesis. Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so \( \aleph_1 \) is not singular - aka [[. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular and a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well.

The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the EBO, since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's.

Strongly inaccessible

Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus epsilon number), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming.

Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). Zermelo referred to the strongly inaccessible cardinals including \( \aleph_0 \) as "Grenzzahlen". However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of uncountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why large cardinals are often called the "higher infinite".

You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.

Grothendieck universes and categoricity

Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory.

A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent.

And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist.