Inaccessible cardinal

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Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy, although weaker notions such as the worldly cardinals can still be viewed as large 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 and so inaccessible cardinals are worldly, but this is not an equivalence.
  • Every inaccessible cardinal $\kappa$ is an aleph fixed point and 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 $\kappa$.
  • Solovay proved that if there is an inaccessible cardinal, then there is an inner model of a forcing extension satisfying ZF+DC in which every set of reals is Lebesgue measurable there. (citation)
  • Shelah proved that Solovay's use of the inaccessible cardinal is necessary, in the sense that in any model of ZF+DC in which every set of reals is Lebesgue measurable, there is an inner model of ZFC with an inaccessible cardinal.
  • Consequently, the consistency of the existence of an inaccessible cardinal with ZFC is equivalent to the impossibility of our constructing a non-measurable set of reals using only ZF+DC.
  • The uncountable Grothedieck universes are precisely the sets of the form $V_\kappa$ for an inaccessible cardinal $\kappa$.
  • The universe axiom is equivalent to the assertion that there is a proper class of inaccessible cardinals.

Contents

Weakly inaccessible cardinal

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.

Levy collapse

The Levy collapse of an inaccessible cardinal $\kappa$ is the $\lt\kappa$-support product of $\text{Coll}(\omega,\gamma)$ for all $\gamma\lt\kappa$. This forcing collapses all cardinals below $\kappa$ to $\omega$, but since it is $\kappa$-c.c., it preserves $\kappa$ itself, and hence ensures $\kappa=\omega_1$ in the forcing extension.

Inaccessible to reals

A cardinal $\kappa$ is inaccessible to reals if it is inaccessible in $L[x]$ for every real $x$. For example, after the Levy collapse of an inaccessible cardinal $\kappa$, which forces $\kappa=\omega_1$ in the extension, the cardinal $\kappa$ is of course no longer inaccessible, but it remains inaccessible to reals.

Universes

When $\kappa$ is inaccessible, then $V_\kappa$ provides a highly natural transitive model of set theory, a universe in which one can view a large part of classical mathematics as taking place. In what appears to be an instance of convergent evolution, the same universe concept arose in category theory out of the desire to provide 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. Namely, 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.

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.

Degrees of inaccessibility

A cardinal $\kappa$ is $1$-inaccessible if it is inaccessible and a limit of inaccessible cardinals. In other words, $\kappa$ is $1$-inaccessible if $\kappa$ is the $\kappa^{\rm th}$ inaccessible cardinal, that is, if $\kappa$ is a fixed point in the enumeration of all inaccessible cardinals. Equivalently, $\kappa$ is $1$-inaccessible if $V_\kappa$ is a universe and satisfies the universe axiom.

More generally, $\kappa$ is $\alpha$-inaccessible if it is inaccessible and for every $\beta\lt\alpha$ it is a limit of $\beta$-inaccessible cardinals.

Hyper-inaccessible

A cardinal $\kappa$ is hyperinaccessible if it is $\kappa$-inaccessible. One may similarly define that $\kappa$ is $\alpha$-hyperinaccessible if it is hyperinaccessible and for every $\beta\lt\alpha$, it is a limit of $\beta$-inaccessible cardinals. Continuing, $\kappa$ is hyperhyperinaccessible if $\kappa$ is $\kappa$-hyperinaccessible.

More generally, $\kappa$ is hyper${}^\alpha$-inaccessible if for every $\beta\lt\kappa$ it is $\kappa$-hyper${}^\beta$-inaccessible.

Every Mahlo cardinal $\kappa$ is hyper${}^\kappa$-inaccessible.

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