Inaccessible

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Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy (although there are some weaker large cardinal notions, such as universe cardinals).

A cardinal $\kappa$ is inaccessible, also equivalently called strongly inaccessible, if it is an uncountable regular strong limit cardinal.

  • If $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC, but this is not an equivalence, since the weaker notion of universe cardinal also have this feature, and are not all regular when they exist.
  • Every inaccessible cardinal $\kappa$ is a beth fixed point, and consequently $V_\kappa=H_\kappa$.
  • (Zermelo) The models of second-order ZFC are precisely the models $\langle V_\kappa,\in\rangle$ for an inaccessible cardinal $\kappa$.
  • The uncountable Grothedieck universes are precisely the sets of the form $V_\kappa$ for an inaccessible cardinal $\kappa$.


Weakly inaccessible

A cardinal $\kappa$ is weakly inaccessible if it is an uncountable regular limit cardinal. Under the GCH, this is equivalent to inaccessibility, since under GCH every limit cardinal is a strong limit cardinal. So the difference between weak and strong inaccessibility only arises when GCH fails badly. Every inaccessible cardinal is weakly inaccessible, but forcing arguments show that any inaccessible cardinal can become a non-inaccessible weakly inaccessible cardinal in a forcing extension, such as after adding an enormous number of Cohen reals (this forcing is c.c.c. and hence preserves all cardinals and cofinalities and hence also all regular limit cardinals). Meanwhile, every weakly inaccessible cardinal is fully inaccessible in any inner model of GCH, since it will remain a regular limit cardinal in that model and hence also be a strong limit there. In particular, every weakly inaccessible cardinal is inaccessible in the constructible universe $L$. Consequently, although the two large cardinal notions are not provably equivalent, they are equiconsistent.


Grothendieck universe

The concept of Grothendieck universes arose in category theory out of the desire to create a hierarchy of notions of smallness, so that one may form such categories as the category of all small groups, or small rings or small categories, without running into the difficulties of Russell's paradox.

A Grothendieck universe is a transitive set $W$ that is closed under pairing, power set and unions. That is,

  • (transitivity) If $b\in a\in W$, then $b\in W$.
  • (pairing) If $a,b\in W$, then $\{a,b\}\in W$.
  • (power set) If $a\in W$, then $P(a)\in W$.
  • (union) If $a\in W$, then $\cup a\in W$.

It follows by a simple inductive argument that the Grothendieck universes are precisely the sets of the form $V_\kappa$, where $\kappa$ is either $0$, $\omega$ or an inaccessible cardinal.


Universe axiom

The Grothendieck universe axiom is the assertion that every set is an element of a Grothendieck universe. This is equivalent to the assertion that the inaccessible cardinals form a proper class.