19/11/27 22:11:18.53 qnEhNItW.net
>>504
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(This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.)
On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately.
More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC),
Von Neumann?Bernays?Godel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.
A formalist might see the meaning[citation needed] of set varying from system to system. Some kinds of Platonists might view particular formal systems as approximating an underlying reality.
Set-theoretic definitions of finiteness
In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S as finite if S admits a bijection to some set of natural numbers of the form {\displaystyle \{x\,|\,x<n\}}{\displaystyle \{x\,|\,x<n\}}.
Mathematicians more typically choose to ground notions of number in set theory, for example they might model natural numbers by the order types of finite well-ordered sets. Such an approach requires a structural definition of finiteness that does not depend on natural numbers.
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