[Math] Euler’s constant: irrationality and proof theory

Question

Let $\gamma$ represent Euler's constant. Is there a real number $x$ such that there is a proof within Zermelo-Fraenkel set theory (ZF) that $x$ is irrational and there is also a proof within ZF that $\gamma + x$ is irrational?

Answer 1

Yes, let $x$ be Chaitin's constant. Then both $x$ and $\gamma + x$ are uncomputable, therefore irrational.