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?

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# Euler's constant: irrationality and proof theory

### 2 Answers

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

Yes, but this is nothing to do with $\gamma$. Let $a$ be any real number. Then there is $x$ so that $x$ and $a+x$ are both irrational. Proof (within ZF): the set of $x$ such that $x$ is rational is countable, the set of $x$ such that $a+x$ is rational is also countable. But $\mathbb R$ is uncountable.

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

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