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.

For an ordinal λ, a cardinal number κ is called λ-shrewd if for every proposition φ using a predicate symbol and with one free variable, and set A ⊆ V_κ with (V_κ+λ, ∈, A) ⊧ φ(κ) there exists an α, λ' < κ with (V_α+λ', ∈, A ∩ V_α) ⊧ φ(α). It is called shrewd if it is λ-shrewd for every λ^[1]^{(Definition 4.1)} (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.^[1]^{(Corollary 4.3)} 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-Π_m-shrewd if for every Π_m proposition φ, and set A ⊆ V_κ with (V_κ+λ, ∈, A) ⊧ φ(κ) there exists an α, λ' < κ with (V_α+λ', ∈, A ∩ V_α) ⊧ φ(α).^[1]^{(Definition 4.1)} Π_m is one of the levels of the Lévy hierarchy, in short one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.

For finite n, an n-Π_m-shrewd cardinals is the same thing as a Π_mⁿ-indescribable cardinal.^{[citation needed]}

If κ is a subtle cardinal, then the set of κ-shrewd cardinals is stationary in κ.^[1]^{(Lemma 4.6)} A cardinal is strongly unfoldable iff it is shrewd.^[2]

λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in that the reflected substructure must be (V_α+λ, ∈, A ∩ V_α), 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

^ ^a ^b ^c ^d M. Rathjen, "The Art of Ordinal Analysis". Accessed June 20, 2022.
^ Lücke, Philipp (2021). "Strong unfoldability, shrewdness and combinatorial consequences". arXiv:2107.12722 [math.LO]. Accessed 4 July 2023.

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). Archived from the original (PDF) on 2009-12-22. Retrieved 2009-08-13.
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, JSTOR 421132, MR 1369172, S2CID 10648711

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[rathjenart-1] M. Rathjen, "The Art of Ordinal Analysis". Accessed June 20, 2022.

[2] Lücke, Philipp (2021). "Strong unfoldability, shrewdness and combinatorial consequences". arXiv:2107.12722 [math.LO]. Accessed 4 July 2023.

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