lets say we have a countable message space for example set of natural numbers then can we have a cryptography system that information-theoretically (Shannon or other equivalence definition) be Perfectly-secret ? if yes please give example if not an intuitionistic or mathematical proof would be great
The answer is "no, it's not possible"
Since this is likely homework, I won't fill in all the details. I will point you to these ideas:
What is the definition of "perfect secrecy". Isn't that the concept that, for a specific ciphertext, each possible plaintext is equiprobable?
It turns that that no probability distribution over a countably infinite set is equidistributed - that there will always be some elements more probable than others.