Can a set containing non-standard elements only, have a legal set formation?

Question

Is it possible to form a set that contains only non-standard elements legally? A legal set formation means that only internal formulas can be used to form the set.

Here's what I thought: A set containing non-standard elements must be a non-standard set. Otherwise, by transfer principle, it has to be empty. Thus it can't be uniquely described by internal mathematics. It can't take the form as $\{x : A(x)\}$.

Mikhail Katz · Accepted Answer · 2013-05-20T12:38:43.733

1

A set containing a single non-standard element is perfectly legal.

If the "standard" predicate cannot be used, then translated into NSA you can only define sets that are natural extensions of real sets $X$. The natural extension of the empty set $X=\emptyset$ is still empty. If $X$ is inhabited, then its natural extension, which contains all the elements of $X$, will necessarily contain standard elements.

edited May 20 '13 at 12:38

answered May 20 '13 at 11:14

Mikhail Katz

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Similarly, an interval both of whose endpoints are unlimited positive numbers is legal but contains no standard points. – Mikhail Katz May 20 '13 at 11:15
Thank you for your answer. Could you please provide an explicit construction, which is only based on axioms of set theory? – Metta World Peace May 20 '13 at 11:18
To define such interval, shouldn't we define the two endpoints first? But I can't come up with a method to define an unlimited number without using the predicate "standard". – Metta World Peace May 20 '13 at 12:29
Thank you. I have to take some time to grok your argument. – Metta World Peace May 20 '13 at 15:25

Can a set containing non-standard elements only, have a legal set formation?

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