Shrewd cardinal
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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, 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[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-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.[1](Definition 4.1) Πm 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-Πm-shrewd cardinals is the same thing as a Πmn-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 (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". http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.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
- ↑ 1.0 1.1 1.2 1.3 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.
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