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By the power of the transfer principle, the principle of internal definition and the overspill principle, most sets in the nonstandard-superstructure behave rather tame (or rather, standard).
However, there are sets that fall outside of the kingdom of these statements, i.e. the external sets that can't be extended to internal sets (e.g. $\{\mathbb{N}, \mathbb{^*N-N} \}$).

As there's nothing restricting their behavior, one would expect them to show all kinds of strange traits.
What are examples for traits of those sets that are strange, i.e. traits which cleary aren't obtained by a transfer of some standard-statement?

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    $\begingroup$ Example: $X$ is a nonempty set of natural numbers with no least element. $\endgroup$ – Alex Kruckman May 24 '18 at 12:51
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    $\begingroup$ Also, just in terms of cardinality, the nonstandard superstructure will have more nonstandard sets than standard ones. $\endgroup$ – Alex Kruckman May 24 '18 at 12:53

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