TITLE: An easy example of a non-constructive proof without an obvious "fix"?
QUESTION [57 upvotes]: I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it.
Euclid's proof of the infinitude of primes came to mind, however there is an obvious way to "fix" it: just try all the numbers between the biggest prime and the constructed number, and you'll find a prime in a finite number of steps.
Are there good examples of simple non-constructive proofs which would require a substantial change to be made constructive? (Or better yet, can't be made constructive at all).

REPLY [5 votes]: There are reasonably-natural games (two player, perfect information) which we can prove are won by Player $1$, but for which no winning strategy is known; or, at least, for which it is not obvious how to transform the classical proof that Player $1$ wins into a constructive proof.
The classic example is Chomp: we start with an $m\times n$ square grid. On each turn, a player removes some tile, along with every tile below and to the left of that tile. The player who takes the top-right square loses.
Since this is a finite game, one player or the other must have a winning strategy. And player $2$ can't have a winning strategy, by a strategy stealing argument: Imagine player $1$ ate the bottom-left square, and player $2$'s winning response was to eat square $\alpha$. Then player $1$ could have just eaten square $\alpha$, and then pretended for the rest of the game that they were player $2$, and force the real player $2$ to lose by playing their own strategy against them.
Although this is perfectly valid classically, it's highly non-constructive.

Actually, determinacy statements in general might push the bounds of constructivity. The usual proof of open determinacy is to assume Open has no winning strategy, and have Closed simply play to preserve Open's lack of a winning strategy. This is impredicative, though (and this is backed up by the fact that computable open games need not have hyperarithmetic winning strategies), and I suspect open determinacy is not true in most constructive theories. (But I could very easily be wrong.)