Editing Inaccessible cardinal (section)

{{Short description|Type of infinite number in set theory}}
{{use dmy dates|date=June 2020}}
In [[set theory]], a [[cardinal number]] is a '''strongly inaccessible cardinal''' if it is [[uncountable set|uncountable]], [[regular cardinal|regular]], and a [[strong limit cardinal]].
A cardinal is a '''weakly inaccessible cardinal''' if it is uncountable, regular, and a [[weak limit cardinal]].

Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it has meant "weakly inaccessible cardinal". Weakly inaccessible cardinals were introduced by {{harvtxt|Hausdorff|1908}}. Strongly inaccessible cardinals were introduced by {{harvtxt|Sierpiński|Tarski|1930}} and {{harvtxt|Zermelo|1930}}; in the latter they were referred to along with <math>\aleph_0</math> as ''Grenzzahlen'' ([[English language|English]] "limit numbers").<ref>A. Kanamori, "[https://math.bu.edu/people/aki/10.pdf Zermelo and Set Theory]", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.</ref>

Every strongly inaccessible cardinal is a weakly inaccessible cardinal.  The [[generalized continuum hypothesis]] implies that all weakly inaccessible cardinals are strongly inaccessible as well.

The two notions of an inaccessible cardinal <math>\kappa</math> describe a cardinality <math>\kappa</math> which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of cardinality less than <math>\kappa</math>.  Hence the word "inaccessible".  By mandating that inaccessible cardinals are uncountable, they turn out to be very large.

In particular, inaccessible cardinals need not exist at all.  That is, it is believed that there are models of [[Zermelo-Fraenkel set theory]], even with the [[axiom of choice]] (ZFC), for which no inaccessible cardinals exist<ref>{{Cite web |last=Joel |first=Hamkins |date=2022-12-24 |title=Does anyone still seriously doubt the consistency of ZFC? |url=https://mathoverflow.net/questions/437195/does-anyone-still-seriously-doubt-the-consistency-of-zfc |publisher=[[MathOverflow]]}}</ref>. On the other hand, it also believed that there are models of ZFC for which even strongly inaccessible cardinals <em>do</em> exist.  That ZFC can accommodate these large sets, but does not necessitate them, provides an introduction to the [[large cardinal|large cardinal axioms]].  See also [[#Models and consistency|Models and consistency]].

The existence of a strongly inaccessible cardinal is equivalent to the existence of a [[Grothendieck universe]].
If <math>\kappa</math> is a strongly inaccessible cardinal then the [[Von Neumann universe|von Neumann stage]] <math>V_{\kappa}</math> is a Grothendieck universe.
Conversely, if <math>U</math> is a Grothendieck universe then there is a strongly inaccessible cardinal <math>\kappa</math> such that <math>V_{\kappa}=U</math>. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.

An [[ordinal number|ordinal]] is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and {{mvar|&omega;}} are regular ordinals, but not limits of regular ordinals.)

From some perspectives, the requirement that a weakly or strongly inaccessible cardinal be uncountable is unnatural or unnecessary. Even though {{tmath|\aleph_0}} is countable, it is regular and is a strong limit cardinal.  {{tmath|\aleph_0}} is also the smallest weak limit regular cardinal.  Assuming the axiom of choice, every other infinite cardinal number is either regular or a weak limit cardinal. However, only a rather large cardinal number can be both.  Since a cardinal {{tmath|\kappa}} larger than {{tmath|\aleph_0}} is necessarily uncountable, if {{tmath|\kappa}} is also regular and a weak limit cardinal then {{tmath|\kappa}} must be a weakly inaccessible cardinal.