Inconsistency of ℕ from a not-finitist point of view

Considering the set of natural numbers ℕ, then in the context of Peano axioms, starting from inequalities between finite sets, we find a fundamental contradiction, about the existence of ℕ, from a not-finitist point of view. 

This proof of inconsistency is not-finitist because it involves infinite totalities.
But this is natural considering set theory with the axiom of infinity and all elements of a set. On the other hand a finitist proof would imply the end of mathematics as we know it. Anyway, refusing a precise definition of N, then refusing the axiom of infinity, could be a view to avoid this inconsistency. So the axiom of infinity would seem to have a similar role to coherence. It is not demonstrable, but also it cannot be taken as an axiom if one doesn't want a system to be inconsistent. This proof supports finitist approach in a not arbitrary manner.