Indescribable cardinal explained

In set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by .

A cardinal number

\kappa is called \Pi ^{n} _{m} -indescribable if for every \Pi _{m} proposition \phi , and set A\subseteq V _{\kappa} with (V _{\kappa+n} ,\in,A)\vDash\phi there exists an \alpha<\kappa with (V _{\alpha+n} ,\in,A\cap V _{\alpha)\vDash\phi} .^[1] Following Lévy's hierarchy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. \Sigma ^{n} _{m} -indescribable cardinals are defined in a similar way, but with an outermost existential quantifier. Prior to defining the structure (V _{\kappa+n} ,\in,A) , one new predicate symbol is added to the language of set theory, which is interpreted as A .^[2] The idea is that \kappa cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties.

The cardinal number

\kappa is called totally indescribable if it is \Pi ^{n} _{m} -indescribable for all positive integers m and n.

If

\alpha is an ordinal, the cardinal number \kappa is called \alpha -indescribable if for every formula \phi and every subset U of V _{\kappa} such that \phi(U) holds in V _{\kappa+\alpha} there is a some λ <\kappa such that \phi(U\cap V _{λ)} holds in V _{λ+\alpha} . If \alpha is infinite then \alpha -indescribable ordinals are totally indescribable, and if \alpha is finite they are the same as \Pi ^{\alpha} _{\omega} -indescribable ordinals. There is no \kappa that is \kappa -indescribable, nor does \alpha -indescribability necessarily imply \beta -indescribability for any \beta<\alpha , but there is an alternative notion of shrewd cardinals that makes sense when \alpha\geq\kappa : if \phi(U,\kappa) holds in V _{\kappa+\alpha} , then there are λ <\kappa and \beta such that \phi(U\cap V _{λ,λ)} holds in V _{λ+\beta} .^[3] However, it is possible that a cardinal \pi is \kappa -indescribable for \kappa much greater than \pi .^{Ch. 9, theorem 4.3}

Historical note

Originally, a cardinal κ was called Q-indescribable if for every Q-formula

\phi and relation A , if (\kappa,<,A)\vDash\phi then there exists an \alpha<\kappa such that (\alpha,\in,A\upharpoonright\alpha)\vDash\phi .^{[4] [5]} Using this definition, \kappa is \Pi ^{1} _{0} -indescribable iff \kappa is regular and greater than \aleph _{0} .^p.207 The cardinals \kappa satisfying the above version based on the cumulative hierarchy were called strongly Q-indescribable.^[6] This property has also been referred to as "ordinal Q -indescribability".^{[7] p.32}

Equivalent conditions

A cardinal is

\Sigma ^{1} _{n +1} -indescribable iff it is \Pi ^{1} _{n} -indescribable.^[8] A cardinal is inaccessible if and only if it is \Pi ^{0} _{n} -indescribable for all positive integers n , equivalently iff it is \Pi ^{0} _{2} -indescribable, equivalently if it is \Sigma ^{1} _{1} -indescribable.\Pi ^{1} _{1} -indescribable cardinals are the same as weakly compact cardinals.

The indescribability condition is equivalent to

V _{\kappa} satisfying the reflection principle (which is provable in ZFC), but extended by allowing higher-order formulae with a second-order free variable.^[8]

For cardinals

\kappa<\theta , say that an elementary embedding j:M\to H(\theta) a small embedding if M is transitive and j( rm{crit}(j))=\kappa . For any natural number 1 \leq n , \kappa is \Pi ^{1} _{n} -indescribable iff there is an \alpha>\kappa such that for all \theta>\alpha there is a small embedding j:M\to H _{\theta} such that H( rm{crit}(j) ^{+)} ^{M\prec} ^{\Sigma _{n}} H( rm{crit}(j) ^{+)} .^{[9] , Corollary 4.3}

If V=L, then for a natural number n>0, an uncountable cardinal is Π-indescribable iff it's (n+1)-stationary.^[10]

Enforceable classes

For a class

X of ordinals and a \Gamma -indescribable cardinal \kappa , X is said to be enforced at \alpha (by some formula \phi of \Gamma ) if there is a \Gamma -formula \phi and an A\subseteq V _{\kappa} such that (V _{\kappa,\in,A)\vDash\phi} , but for no \beta<\alpha with \beta\notin X does (V _{\beta,\in,A\cap} V _{\beta)\vDash\phi} hold.^p.277 This gives a tool to show necessary properties of indescribable cardinals.

Properties

The property of

\kappa being \Pi ^{1} _{n} -indescribable is \Pi ^{1} _{n +1} over V _{\kappa} , i.e. there is a \Pi ^{1} _{n +1} sentence that V _{\kappa} satisfies iff \kappa is \Pi ^{1} _{n} -indescribable.^[11] For m> 1 , the property of being \Pi ^{m} _{n} -indescribable is \Sigma ^{m} _{n} and the property of being \Sigma ^{m} _{n} -indescribable is \Pi ^{m} _{n} . Thus, for m> 1 , every cardinal that is either \Pi ^{m} _{n +1} -indescribable or \Sigma ^{m} _{n +1} -indescribable is both \Pi ^{m} _{n} -indescribable and \Sigma ^{m} _{n} -indescribable and the set of such cardinals below it is stationary. The consistency strength of \Sigma ^{m} _{n} -indescribable cardinals is below that of \Pi ^{m} _{n} -indescribable, but for m> 1 it is consistent with ZFC that the least \Sigma ^{m} _{n} -indescribable exists and is above the least \Pi ^{m} _{n} -indescribable cardinal (this is proved from consistency of ZFC with \Pi ^{m} _{n} -indescribable cardinal and a \Sigma ^{m} _{n} -indescribable cardinal above it).

Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for

\Pi ^{m} _{n} - and \Sigma ^{m} _{n} -indescribability.

For natural number

n , if a cardinal \kappa is n -indescribable, there is an ordinal \alpha<\kappa such that (V _{\alpha+n} ,\in)\equiv(V _{\kappa+n} ,\in) , where \equiv denotes elementary equivalence.^[12] For n =0 this is a biconditional (see Two model-theoretic characterisations of inaccessibility).

Measurable cardinals are

\Pi ^{2} _{1} -indescribable, but the smallest measurable cardinal is not \Sigma ^{2} _{1} -indescribable. However, assuming choice, there are many totally indescribable cardinals below any measurable cardinal.

For

n\geq 1 , ZFC+"there is a \Sigma ^{1} _{n} -indescribable cardinal" is equiconsistent with ZFC+"there is a \Sigma ^{1} _{n} -indescribable cardinal \kappa such that 2 ^{\kappa>\kappa} ^{+} ", i.e. "GCH fails at a \Sigma ^{1} _{n} -indescribable cardinal".^[8]

References

• • Book: Kanamori, Akihiro. Akihiro Kanamori. 2003. Springer. The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite. 2nd. 3-540-00384-3. 10.1007/978-3-540-88867-3_2.

Citations

Notes and References

1. Book: Drake, F. R.. Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. 1974. 0-444-10535-2.
2. Book: Jech . Thomas. Set Theory: The Third Millennium Edition, revised and expanded. Springer Monographs in Mathematics. 2006. 3-540-44085-2. 10.1007/3-540-44761-X. 295.
3. M. Rathjen, "The Higher Infinite in Proof Theory" (1995), p.20. Archived 14 January 2024.
4. K. Kunen, "Indescribability and the Continuum" (1971). Appearing in Axiomatic Set Theory: Proceedings of Symposia in Pure Mathematics, vol. 13 part 1, pp.199--203
5. Azriel Lévy, "The Sizes of the Indescribable Cardinals" (1971). Appearing in Axiomatic Set Theory: Proceedings of Symposia in Pure Mathematics, vol. 13 part 1, pp.205--218
6. Richter . Wayne. Aczel . Peter. Inductive Definitions and Reflecting Properties of Admissible Ordinals. 1974. Studies in Logic and the Foundations of Mathematics. 79. 301–381. 10.1016/S0049-237X(08)70592-5. 10852/44063. free.
7. W. Boos, "Lectures on large cardinal axioms". In Logic Conference, Kiel 1974. Lecture Notes in Mathematics 499 (1975).
8. Hauser . Kai. Indescribable Cardinals and Elementary Embeddings. 2274692. Journal of Symbolic Logic. 56. 2. 1991. 439–457. 10.2307/2274692.
9. Holy . Peter. Lücke . Philipp. Njegomir . Ana. Small embedding characterizations for large cardinals. Annals of Pure and Applied Logic. 170. 2. 2019. 251–271. 10.1016/j.apal.2018.10.002 . free. 1708.06103.
10. Bagaria . Joan. Magidor . Menachem . Menachem Magidor. Sakai . Hiroshi. Reflection and indescribability in the constructible universe. Israel Journal of Mathematics. 208. 1–11. 2015. 10.1007/s11856-015-1191-7.
11. Book: Kanamori, Akihiro. Akihiro Kanamori. 2003. Springer. The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite. 2nd. 3-540-00384-3. 10.1007/978-3-540-88867-3_2. 64.
12. W. N. Reinhardt, "Ackermann's set theory equals ZF", pp.234--235. Annals of Mathematical Logic vol. 2, iss. 2 (1970).