Is it possible to express well-ordering in a first-order language?(adsbygoogle = window.adsbygoogle || []).push({});

For example:

If [itex]X[/itex] is a set and [itex]<[/itex] is a binary relation on [itex]X[/itex] such that [itex](X,<) \vDash \forall x \; \neg(x<x)[/itex] and [itex](X,<) \vDash \forall x \forall y \forall z((x<y \wedge y<z) \rightarrow x<z)[/itex], then [itex](X,<)[/itex] is a partial order.

If [itex](X,<)[/itex] is a partial order such that [itex](X,<) \vDash \forall x \forall y(x<y \vee x=y \vee y<x)[/itex], then [itex](X,<)[/itex] is a total order.

I cannot think of a way to express well-ordering in this language. Since the quantifiers range over all of [itex]X[/itex] and there is no obvious way to quantify over subsets of [itex]X[/itex], I am thinking it is not possible to express well-ordering in this way. But it is also possible that I am just not sufficiently clever to think of something.

Any help?

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# Well-ordering in a first-order language

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