# Are there sets $S\subseteq\Bbb N$ which are provably non-empty, but we don't know what is $\min S$? [duplicate]

I was wondering if there is a property that is known to be satisfied for certain "things" but for which we do not know any explicit example.

More explicitly (also more restrictive but possibly easier):

Is there a set $S \subseteq \mathbb{N}$ for which one can prove that $S \neq \varnothing$ and yet there is no known (to us, humans, at this point in time) element of $S$?

• Plenty of singleton sets here: math.stackexchange.com/questions/1315615/… – JP McCarthy Jul 21 '15 at 18:29
• Every theorem that states the existence of some constant in a non-effective way would qualify. For example (though in $\Bbb{R}$, and not in $\Bbb{N}$) in 2002 Becher and Figueira showed the existence of some computable normal numbers. While they do provide an algorithm, it is so inefficient that no digits of any of those numbers are known. – A.P. Jul 21 '15 at 18:33
• @Pedro: I don't know if this is really about constructive mathematics. – Asaf Karagila Jul 21 '15 at 18:35
• @AsafKaragila I voted to reopen... this is not a duplicate of that question... that question is arguably a subquestion of this one. – JP McCarthy Jul 21 '15 at 18:37
• We know that non-measurable sets exist, but they are not constructable. – Alex S Jul 21 '15 at 18:38