Editing Internal set theory (section)

==== Applied to the relation ≠ ====
If ''S'' is standard and finite, we take for the relation {{tmath|R(g, f)}}: ''g'' and ''f'' are not equal and ''g'' is in ''S''.  Since "''For every standard finite set F there is an element g in S such that {{tmath|g \ne f}} for all f in {{itco|F}}''" is false (no such ''g'' exists when {{math|''F'' {{=}} ''S''}}), we may use Idealisation to tell us that "''There is a G in S such that {{tmath|g \ne f}} for all standard {{itco|f}}''" is also false, i.e. all the elements of ''S'' are standard.

If ''S'' is infinite, then we take for the relation {{tmath|R(g, f)}}: ''g'' and ''f'' are not equal and ''g'' is in ''S''. Since "''For every standard finite set F there is an element g in S such that {{tmath|g \ne f}} for all f in {{itco|F}}''" (the infinite set ''S'' is not a subset of the finite set {{itco|''F''}}), we may use Idealisation to derive "''There is a G in S such that {{tmath|g \ne f}} for all standard {{itco|f}}''."  In other words, every infinite set contains a nonstandard element (many, in fact).

The power set of a standard finite set is standard (by Transfer) and finite, so all the subsets of a standard finite set are standard.

If ''S'' is nonstandard, we take for the relation {{tmath|R(g, f)}}: ''g'' and ''f'' are not equal and ''g'' is in ''S''. Since "''For every standard finite set F there is an element g in S such that {{tmath|g \ne f}} for all f in {{itco|F}}''" (the nonstandard set ''S'' is not a subset of the standard and finite set {{itco|''F''}}), we may use Idealisation to derive "''There is a G in S such that {{tmath|g \ne f}} for all standard f.''"  In other words, every nonstandard set contains a nonstandard element.

As a consequence of all these results, all the elements of a set ''S'' are standard if and only if ''S'' is standard and finite.