can all the math problems be solved in an infinity but countable time assume that we were inmortals so we can past an infinity amount of time doing calculations with the aid of a computer
then assume an hypothetitc inmortal being who wants to solve Riemann Hypothesis
then if he needs 10 seconds to evaluate a Riemann zero , after a countable (but infinite) amount of time he would know that all the riemann zeros has real part 1/2
the same for fermat theorem $$ x^{p}+y^{p}=z^{p} $$
then for any fixed prime p the robot would evaluate if the above equation has a solution
my question is does it happen (hypothetically) to every math problem??
can ANY math problem be solved in an infinite amount of time if we had the ability to live forever?=
 A: This question can be made precise via mathematical logic, specifically computability theory and the arithmetic hierarchy: the problems which are resolvable in countable time in the most naive sense - one moment for each natural number - are exactly those which are $\Pi^0_1$ or $\Sigma^0_1$, that is, which assert the nonexistence or existence, respectively, of a finite sequence of natural numbers with some "checkable" property; e.g., being a counterexample to Fermat is easily checkable. It is known that the arithmetic hierarchy is strict, so that there are problems (just about natural numbers!) which cannot be resolved in the manner you describe.
There are lots of problems which appear not of this form, but are equivalent to a statement of this form; for instance, the Riemann hypothesis (mentioned in the comments) is on the face of it a statement about all real numbers, of which there are uncountably many, but turns out to be equivalent to a $\Pi^0_1$ sentence.

However, "countable time" is a bit more flexible than you've described. We can generally talk about more complicated infinitary proofs, and more complicated infinite time setups. For example, the statement "every digit occurs infinitely often in $\pi$" is perhaps surprisingly not (known to be) equivalent to a $\Pi^0_1$ or $\Sigma^0_1$ sentence; however, it can be verified/disproved in countable time as follows: for each $n$, the question "each digit appears at least $n$ times" can be verified/disproved in infinite time in the manner of your question, and so the whole statement can be verified/disproved in "infinitely many infinite steps" - or, time $\omega^2$. This is a bit vague, but it can be made rigorous without too much work.
The various generalizations of infinitary proofs and computations have been studied extensively in logic, throughout computability theory, proof theory, and set theory. There are far too many things to link to, but here are some things you might be interested in:


*

*The $\omega$-rule (or $\Omega$-rule), an infinitary proof rule for constructing proofs of the kind you describe (and much more).

*Ordinal analysis, the study of what kinds of infinitary methods are required to prove various theorems; often connected with infinitary proofs and the $\omega$-rule (or its fragments/extensions) in particular.

*The hyperarithmetic hierarchy, extending the notion of "countable proof" a huge way (and higher recursion theory in general, and arguably fine structure theory if we squint). 

*Infinite-time Turing machines and ordinal Turing machines, exactly what they sound like.
