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つづき
（参考：Second-order の無限公理）
URLﾘﾝｸ(plato.stanford.edu)
Stanford Encyclopedia of Philosophy
Second-order and Higher-order Logic
First published Thu Aug 1, 2019 by Jouko Väänänen
1. Introduction
2. The Syntax of Second-Order Logic
3. The Semantics of Second-Order Logic
3.1 The Ehrenfeucht-Fraïssé game of second-order logic
4. Properties of Second-Order Formulas
5. The Infamous Power of Second-Order Logic
5.1 Putting distance between second- and first- order logic
5.2 The collapse of the Completeness Theorem
5.3 “Set theory in sheep’s clothing”
5.4 Does second-order logic depend on the Axiom of Choice?
6. Non-Absoluteness of Truth in Second-Order Logic
7. Model Theory of Second-Order Logic
7.1 Second-order characterizable structures
7.2 Second-order logic and large cardinals
7.3 The model theory of general and Henkin models
8. Decidability Results
9. Axioms of Second-Order Logic
9.1 General models and Henkin models
9.2 Axioms of infinity
10. Categoricity
11. Logics Between First and Second Order
12. Higher Order Logic vis-à-vis Type Theory
13. Foundations of Mathematics
14. Second-Order Arithmetic
15. Second-Order Set Theory
16. Finite Model Theory
9.2 Axioms of infinity
Some are equivalent if the Axiom of Choice is assumed. Let us call a second-order sentence
ϕ of the empty vocabulary an Axiom of Infinity if
A |-ϕ if and only if A is infinite.
An axiom of infinity can say in second-order logic that a proper subset of the domain has the same cardinality as the entire domain (i.e., that the domain is not Dedekind-finite), or that there is a partial order without a maximal element, or that there is a set with a unary function and a constant which constitute a structure isomorphic to (N,s,0), or that the domain is the union of two disjoint sets which have the same cardinality as the domain, and so on. As is the case in set theory without the Axiom of Choice, the different formulations of infiniteness need not be equivalent. In second-order logic the situation is even more diffuse because of the variety of different formulations of the Axiom of Choice. We refer to Asser (1981) for a discussion of the different variants and to Hasenjaeger (1961) for a proof that the various non-equivalent forms of Axioms of Infinity form in a sense a dense set. For a survey of different concepts of finiteness, see de la Cruz (2002).
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