TITLE: Existential statement without witness
QUESTION [12 upvotes]: Are there existential theorems of ZFC, or PA say, with no witnesses?
Ie does there exist a formula $\phi$ such that ZFC $\vdash\exists x \phi(x)$, but for all numerals $\underline{n}$, ZFC $\nvdash \phi(\underline{n})$?
Can you give an example of such a formula?

REPLY [10 votes]: $\def\epn{\mathrm{EP}_\mathbb N}\def\dp{\mathrm{DP}}$When a theory $T$ (with a distinguished definable set of “natural numbers” satisfying a suitable minimal arithmetic) has the property in the question, viz.
$$\tag{$\epn$}T\vdash\exists x\in\mathbb N\,\phi(x)\text{ implies }T\vdash\phi(\underline n)\text{ for some $n\in\mathbb N$,}$$
$T$ is said to have the numerical existence property. As explained in Joel’s answer, classical theories can have the numerical existence property only if inconsistent or complete, which prevents any interesting examples by virtue of Gödel’s incompleteness theorem.
However, the situation is quite different for theories over intuitionistic logic. Indeed, most natural constructive theories (e.g., Heyting arithmetic) do have $\epn$.
Since $\phi\lor\psi$ is equivalent to $\exists n\,((n=0\land\phi)\lor(n=1\land\psi))$, $\epn$ implies the seemingly weaker disjunction property
$$\tag{$\dp$}T\vdash\phi\lor\psi\implies T\vdash\phi\text{ or }T\vdash\psi.$$
By a result of Harvey Friedman, $\dp$ is in fact equivalent to $\epn$ if $T$ is recursively axiomatizable.
We may stratify the definition by restricting the complexity of the formulas. A consistent recursively axiomatizable classical theory never has $\epn$ for $\Pi^0_1$ formulas, or $\dp$ for $\Pi^0_1$ sentences, or $\dp$ for $\phi\in\Sigma^0_1$ and $\psi\in\Pi^0_1$. This leaves $\Sigma^0_1$ as the only realistic possibility. Friedman’s argument shows that for recursively axiomatized $T$, the following are equivalent:

$T$ has $\epn$ for $\Sigma^0_1$ formulas,
$T$ has $\dp$ for $\Sigma^0_1$ sentences,
$T$ is $\Sigma^0_1$-sound or inconsistent.

Naturally occurring theories are $\Sigma^0_1$-sound (e.g., PA) or assumed to be $\Sigma^0_1$-sound (e.g. ZFC).
See also Visser (§7) for an interesting connection of Friedman’s characterization to (non)interpretability of inconsistency.