Re: Problems in NP with infinite number of possibiltiies to check?

No.

The additional input to the P verifier (certificate or proof) is
(polynomially) bounded, so you can have only a finite number of
different 'certificates' to explore. In other words, the verifier can
only read a finite number of bits in its time bound.

If for the P verifier time-bounded by n^k the the number of
possibilities to check is bounded by 2^(n^k)

.