Euler's constant: irrationality and proof theory 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?
 A: I got the feeling that maybe what the OP wanted was: Provide an   explicit constructive example   of an irrational $x$ such that $x+\gamma$ is irrational. In any case, under this respect the question seems interesting as well, so I would like to consider it. 
While the existence is quite clear (e.g. there exists $x\in\{\sqrt{2},2\sqrt{2} \}$ with this property, and also  $ x\in\{\sqrt{2},\gamma \}$), giving an explicit example is a  bit less obvious. 
As suggested in the various answers, we may consider the problem   more in general  than for the case where $\gamma$ is the Euler-Mascheroni constant. I'd like to formalize the question in   the following Problem:

Split a given  computable  real number $\gamma$ into a sum of two irrational computable real numbers.

There is a cute construction that uses a  "balanced  factorial representation" of real numbers. Recall that any real number $\gamma$
can be written in the form 
$$\gamma=\sum_{k\ge 1}\frac{\gamma_k}{k!}$$
with coefficients $\gamma_k\in\mathbb{Z}$ verifying 
$$|\gamma_k |\le k-1 \ ,$$
for any $k\ge 2$. A number of this form is rational if and only if the sequence $\gamma_k$ is either eventually equal to  $0$, or eventually equal to $k-1$, or eventually equal to $-(k-1)$. Moreover,   $\gamma$ is a computable number  if and only if the coefficients $\gamma_k$ are given by some computable function. (Note: as remarked in comments, this is generally not true for  base systems with nonnegative coefficients: the binary expansion of a computable number may not be computable, which reflects the fact that one may not be able to decide whether $\gamma$ is larger or smaller than a given rational approximation of its, no matter how good). 
Given the  two facts above, the problem is then translated into: split computably the sequence $\gamma_k$  as a sum $ \gamma_k=\alpha_k+\beta_k$, in such a way that $$ 0<|\alpha_k|<k-1$$ and $$ 0<|\beta_k|<k-1$$
for infinitely many $k$. Again, changing sign coefficients make everything quite easy to deal with: we can e.g. eventually take  both $|\alpha_k|$ and $|\beta_k|$ in the interval $[k/4,3k/4]$. 
A: Yes, let $x$ be Chaitin's constant. Then both $x$ and $\gamma + x$ are uncomputable, therefore irrational.
A: 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.
