Inaccessible cardinals

**Inaccessible cardinals** are cardinals that are both limit cardinals and regular cardinals. The ath inaccessible is commonly denoted Ia and the first, I1, is usually just written as I.

## Definition

There are two definitions of inaccessible cardinals depending on whether one chooses *weak* limit cardinals (cardinals of the form ℵa where a is a limit ordinal) versus *strong* limit cardinals (ℶa where a is a limit ordinal). These are termed the "weakly inaccessibles" and the "strongly inaccessibles" respectively. They are the same assuming the generalized continuum hypothesis. If left unspecified it is assumed the strongly inaccessibles are being talked about.

It has been shown by Edwin Shade that it takes at most 37,915 symbols under a language L = {¬,∃,∈,xn} to assert the existence of the first inaccessible cardinal. This likely means that ZFC + "There exists an inaccessible cardinal" is many times the size of ZFC when comapring the symbol count of both theories' base axioms.

The following are equivalent definitions of inaccessible cardinals:

* Regular limits of regular cardinals.
* Limit cardinals that are also regular, including the subclass of regular worldly cardinals.
* Regular fixed points of omega fixed points f(α) = Ωα.

Note the concept of regularity is involved in all three definitions. This becomes a defining property which is diagonalized over in the definition of a Mahlo cardinal.

## Relevance in model theory

The first inaccessible cardinal I can be used to model the universe of all sets which can be defined in ZFC. VI, the first inaccessible stage of the Von Neumann hierarchy, contains all sets realizable in ZFC and thus satisfies every provable ZFC theorem. "ZFC + VI exists" is in turn modeled by VI2. The class of strongly inaccessibles almost exactly coincides with the class of Grothendieck universes due to the property that Vk is transitive and closed under the powersetting, replacement, and pairing axioms of ZFC when k is strongly inaccessible. (The classes are not exactly the same though since the null set and the set of finite hereditary definable sets are counted as Grothendieck universes but not as strongly inaccessibles.)

## Relevance in googology

Inaccessible cardinals are most often used in ordinal collapsing functions, where their ordinal forms are folded down into large countable ordinals with well defined fundamental sequences which can be used in conjunction with ordinal hierarchies such as the FGH, HGH, and SGH.

## Extensions

Just as inaccessibles are formed from the intersection of limit cardinals with regular cardinals, we can form a fundamentally higher type of inaccessible called a *1-inaccessible* from the intersection of inaccessibles with limits of inaccessibles. It is important to note that the limit of an inaccessible is not always inaccessible itself, as I1, I2, I3,. . . forms a length-ω fundamental sequence to a limit cardinal which by definition can't be regular. Thus Iω is much larger.

In general, a-inaccessibles are formed from the intersection of all b-inaccessibles with limits of b-inaccessibles where b < a. We may go even further by defining *hyper-inaccessibles*, which are b such that b is b-inaccessible. Just as there are ordinal collapsing functions to convert inaccessibles into large countable ordinals, there are collapsing extensions which turn Mahlo ordinals into inaccessibles, hyper-inaccessibles, and so forth.

1. ↑ Shade, Edwin. (1 June 2021) How To Rayo-Name Large Cardinals. *Googology Testing Wiki*. File:Rayo-Naming Inaccessibles.pdf Retrieved 17 October 2021.