Consider $\phi(A)$ a formula of second-order arithmetic with one free variable $A$ of type "set". Suppose $\exists A : \phi(A)$ is a true sentence. Does it follow (not in second order arithmetic itself, but in a stronger theory of your choice, e.g. ZFC) that there is a formula of second-order arithmetic $\psi(n)$ with one free variable $n$ of type "number", such that $\phi(\lbrace n | \psi(n)\rbrace)$ is a true sentence?



Are there "non-constructive" sets in second-order arithmetic?
This may or may not be true depending on the background set theory.


On the one hand, it is consistent with ZFC that there exists a $\Delta^1\_2$-definable well-ordering $\prec$ of the reals. Then one can take for $\psi(n)$ the formula saying “$n$ is in the $\prec$-minimal set $A$ satisfying $\phi(A)$”, so the answer is positive.


On the other hand, it is also consistent with ZFC that the answer is negative. For example, assume that $V$ is the Cohen forcing extension of $L$, and take for $\phi(A)$ the $\Pi^1\_2$ formula saying “$A$ is not constructible”. Then $\exists A\,\phi(A)$ is true, but since Cohen forcing is homogeneous, every parameter-free definable set (not just in the language of second-order arithmetic, but in full language of ZFC) is in the ground model, which contains no witnesses to $\phi(A)$.


On the third hand, ZFC proves the answer is positive when $\phi(A)$ is $\Sigma^1\_2$: if there exists an $A$ such that $\phi(A)$, there also exists such an $A$ in $L$ by Shoenfield absoluteness, and we can define a particular one using a well-ordering of constructible reals as in the first part of the answer.


