25/11/05 11:15:26.89 K/Lr81ky.net
>>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)
URLﾘﾝｸ(www.aichi-gakuin.ac.jp)
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＞
URLﾘﾝｸ(arxiv.org)
[Submitted on 29 Aug 2023]
Conservativity of Type Theory over Higher-order Arithmetic
Benno van den Berg, Daniël Otten
URLﾘﾝｸ(en.wikipedia.org)
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]
URLﾘﾝｸ(mathoverflow.net)
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