[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 77 (1002ﾚｽ)

Inter-universal geometry と ABC予想 (応援スレ) 77 http://rio2016.5ch.io/test/read.cgi/math/1761878205/

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253: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/11/05(水) 11:15:26.89 ID:K/Lr81ky >>252 補足 用語について ・高階算術 ・finite order arithmetic ・Higher-order logic ・Higher order arithmetic by Colin McLarty自身 2014 mathoverflow (参考) http://www.aichi-gakuin.ac.jp/~yamasita/slides/wakate_izawa.pdf 高階算術における抽象論 井澤　昇平 東北大学理学研究科数学専攻 数学基礎論若手の会’１０ Contents. 1 Introduction. 2 Definition of axioms of finite order arithmetic. 検索＜higher order arithmetic＞ https://arxiv.org/abs/2308.15288 [Submitted on 29 Aug 2023] Conservativity of Type Theory over Higher-order Arithmetic Benno van den Berg, Daniël Otten https://en.wikipedia.org/wiki/Higher-order_logic Higher-order logic In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic. The term "higher-order logic" is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types. Leon Chwistek and Frank P. Ramsey proposed this as a simplification of ramified theory of types specified in the Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Simple types is sometimes also meant to exclude polymorphic and dependent types.[1] https://mathoverflow.net/questions/190477/higher-order-arithmetic-and-fragments-of-zfc Higher order arithmetic and fragments of ZFC Zbierski "Models for Higher Order Arithmetics" (BULL. DE L'ACAD. POLONAISE DES SCIENCES Serie des sciences math., astr. et phys. - Vol. XIX, No. 7, 1971) defines ZFn as ZFC with the power set axiom limited to n successive power sets starting from the natural numbers N . Note this includes the axiom of choice. He proves ZFn is a conservative extension of n+2 order arithmetic with the axiom scheme of choice.. He notes Gandy had proved already ZF2 is not conservative over 4 th order arithmetic without the axiom scheme of choice. But what is known about the case where the axiom of choice is also removed from ZF2 ? Sochor "Constructibility in higher order arithmetics" (Arch. Math. Logic (1993) 32:381-389) shows that n -th order arithmetic without choice can interpret n -th order arithmetic with choice.though obviously it is not a conservative extension. So the issue is not equiconsistency. It is conservativity. asked Dec 11, 2014 at 13:57 Colin McLarty http://rio2016.5ch.io/test/read.cgi/math/1761878205/253

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