I have read, that properties of sets such as that every subset has supremum or that set is well ordered cannot be expressed in the language of first-order logic. Well, when I tried to write these things, I seemed to write them in first order language, which really bothers me. So please, tell me, where is the point, where I use something such as quantifying over predicates:(adsbygoogle = window.adsbygoogle || []).push({});

A is well ordered set iff:

[tex](\forall s)((s\,\text{is subset of}\,A)\rightarrow(\exists y)(y\,\text{is least element of}\,s))[/tex]

and (s is subset of A) can be writen as well formed first-order formula with free variables s and A: [tex](\forall x)(x\in s\rightarrow x\in A)[/tex]

and (y is least element of s) can be writen as well formed first-order formula with free variables s and y: [tex]((\forall x)(x\in s\rightarrow y\leq x))\,\&\,(y\in s)[/tex].

I think, that written formulla for definition of well ordered set is well formed first-order formula with one free variable A.

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# Second-order logic

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