Let us assume there is a number which is well-defined and computable, but it is hard to compute it. E.g., $x=\pi^{(\pi^{(\pi^\pi)})}$. It is not even known if the given number is an integer (which is actually true for the chosen $x$). And it is not known if the expression is equal to a prime number (a prime of $\mathbb{N}$).
My question is now: Can it be the case that it is undecidable (in ZFC) if the given expression for $x$ is equal to a prime number? Or is this for all (computable) expressions deciable?
If someone computes this expression to a given accuracy it might be de the case that it ends in something like "...2113.12131...". Then it obviously is not an integer (and therefore not a prime). But what if the expression ends (given any accuracy we choose during the experiments) with ".2113.000000..." or with "...2112.99999...". Then it might be really hard to prove that the expression is an integer (and then maybe a prime number) or not. Can it be that it is not even deciable? Even if the expression itself is computable?
Thank you
First: If it's decidable whether it's an integer, it's decidable whether it's prime - just compute the number to a high enough precision to decide which integer it is, and check whether that's prime.
Second: It's not decidable whether a given expression encodes an integer. Let $x = \sum_{n = 1}^{\infty}p(n)2^{-n}$, where $p(n)$ is defined to be $1$ if $n$ is the Godel number of a proof in ZFC of $1 = 0$ and $0$ otherwise. Clearly, $x$ is $0$ if ZFC is consistent, and is some number strictly between $0$ and $1$ otherwise. Since ZFC can't prove whether ZFC is consistent (assuming, for the moment, that ZFC actually is consistent) ZFC cannot prove whether $x$ is an integer.