Shrewd cardinal

In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by (Rathjen 1995)., extending the definition of indescribable cardinals.

A cardinal number κ is called λ-shrewd if for every proposition φ, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ. It is called shrewd if it is λ-shrewd for every λ (including λ > κ).

This definition extends the concept of indescribability to transfinite levels. A λ-shrewd cardinal is also μ-shrewd for any ordinal μ < λ. Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of Π12-comprehension. It is essentially the nonrecursive analog to the stability property for admissible ordinals.

More generally, a cardinal number κ is called λ-Πm-shrewd if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ.

Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.

For finite n, an n-Πm-shrewd cardinals is the same thing as a Πmn-indescribable cardinal.

If κ is a subtle cardinal, then the set of κ-shrewd cardinals is stationary in κ. Rathjen does not state how shrewd cardinals compare to unfoldable cardinals, however.

λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in that the reflected substructure must be (Vα+λ, ∈, A ∩ Vα), making it impossible for a cardinal κ to be κ-indescribable. Also, the monotonicity property is lost: a λ-indescribable cardinal may fail to be α-indescribable for some ordinal α < λ.

References
  • Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
  • Rathjen, Michael (2006). "The Art of Ordinal Analysis" (PDF).
  • Rathjen, Michael (1995), "Recent advances in ordinal analysis: Π12-CA and related systems", The Bulletin of Symbolic Logic, 1 (4): 468–485, doi:10.2307/421132, ISSN 1079-8986, MR 1369172


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