I have a question regarding the validity of infinity in the logic of math.
If we can define logically a number by it's infinite sum (as a limit of series ...etc) than how can it be logic to use the infinite decent as a proof for non existence of solution (like in Fermat proof for degree 3.
If is it OK to consider the infinite limit as a contradiction than how can we accept the irrational numbers as an infinite decimal expansion. I hope I was clear in my question at least...
What is behind any 'infinite descent' argument is a so-called well-order.
The poster child of an infinite well-order is $(\mathbb N; <)$, i.e. the set of natural numbers with their natural order. The key aspect of a well-order $(X; \prec)$ is that there are no infinite decreasing sequences, i.e. there is no sequence $(x_n \mid n \in \mathbb N)$ such that for all $n \in \mathbb N$
If you can show that a certain order is a well-order and that the contrapositive of your claim yields an infinite decreasing sequence in said order, you derive the desired contradiction. This sort of argument is sometimes called 'proof by infinite descent'. (Which, in my opinion, is a poor choice of words.)