Editing Inaccessible cardinal (section)

== Two model-theoretic characterisations of inaccessibility ==
Firstly, a cardinal {{mvar|κ}} is inaccessible if and only if {{mvar|κ}} has the following [[Reflection principle|reflection]] property: for all subsets <math>U\subset V_\kappa</math>, there exists <math>\alpha<\kappa</math> such that <math>(V_\alpha,\in,U\cap V_\alpha)</math> is an [[elementary substructure]] of <math>(V_\kappa,\in,U)</math>. (In fact, the set of such ''α'' is [[Club set|closed unbounded]] in {{mvar|κ}}.) Therefore, <math>\kappa</math> is <math>\Pi_n^0</math>-[[Totally indescribable cardinal|indescribable]] for all ''n'' ≥ 0. On the other hand, there is not necessarily an ordinal <math>\alpha>\kappa</math> such that <math>V_\kappa</math>, and if this holds, then <math>\kappa</math> must be the <math>\kappa</math>th inaccessible cardinal.<ref>A. Enayat, "Analogues of the MacDowell-Specker_theorem for set theory" (2020), p.10. Accessed 9 March 2024.</ref>

It is provable in ZF that <math>V</math> has a somewhat weaker reflection property, where the substructure <math>(V_\alpha,\in,U\cap V_\alpha)</math> is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation {{math|⊧}} can be defined, semantic truth itself (i.e. <math>\vDash_V</math>) cannot, due to [[Tarski's undefinability theorem|Tarski's theorem]].

Secondly, under ZFC [[Zermelo's categoricity theorem]] can be shown, which states that <math>\kappa</math> is inaccessible if and only if <math>(V_\kappa,\in)</math> is a model of [[Second order logic|second order]] ZFC.

In this case, by the reflection property above, there exists <math>\alpha<\kappa</math> such that <math>(V_\alpha,\in)</math> is a standard model of ([[First order logic|first order]]) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a transitive model of ZFC.

Inaccessibility of <math>\kappa</math> is a <math>\Pi^1_1</math> property over <math>V_\kappa</math>,<ref>K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of Symbolic Logic vol. 56, iss. 2 (1991), pp.439--457.</ref> while a cardinal <math>\pi</math> being inaccessible (in some given model of <math>\mathrm{ZF}</math> containing <math>\pi</math>) is <math>\Pi_1</math>.<ref>K. J. Devlin, "Indescribability Properties and Small Large Cardinals" (1974). In ''<math>\vDash</math>ISILC Logic Conference: Proceedings of the International Summer Institute and Logic Colloquium, Kiel 1974'', Lecture Notes in Mathematics, vol. 499 (1974)</ref>