Let $S$ be a first order definable Martin-Löf random set such as Chaitin's $\Omega$. If Peano Arithmetic, or ZFC, or any other theory with a computable set of axioms, proves infinitely many facts of the form $n\in S$ or $n\not\in S$ then it follows that $S$ is not immune or not co-immune and hence not ML-random after all.

(The set of theorems of our theory of the form $n\in S$ (or $n\not\in S$) is computably enumerable and infinite, hence has an infinite computable subset. Being immune means having no infinite computable subset.)

So only finitely many such facts can be proved. Now using an effective bijection between $\mathbb N$ and $\mathbb N\times \mathbb N$, decompose $S$ into infinitely many "columns", $S=S_0\oplus S_1\oplus\cdots$. Then one of these columns has the required property.

The theory should be strong enough to deal effectively with breaking a definable set up into columns and associating values in a column with values in the original set, but this is certainly doable in PA or ZFC.

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Let $S$ be a first order definable Martin-Löf random set such as Chaitin's $\Omega$. If Peano Arithmetic, or ZFC, or any other theory with a computable set of axioms, proves infinitely many facts of the form $n\in S$ or $n\not\in S$ then it follows that $S$ is not immune or not co-immune and hence not ML-random after all.

(The set of theorems of our theory is computably enumerable and infinite, hence has an infinite computable subset. Being immune means having no infinite computable subset.)

So only finitely many such facts can be proved. Now using an effective bijection between $\mathbb N$ and $\mathbb N\times \mathbb N$, decompose $S$ into infinitely many "columns", $S=S_0\oplus S_1\oplus\cdots$. Then one of these columns has the required property.