Why do we write 'finite' as an adjective to proposition when defining an axiomatic system? At 58:54, Professor Frederic Schuller defines an axiomatic system in the following way:

An axiomatic system is a finite sequence of propositions $a_1,a_2....,a_n$ which are called axioms

Why do we have the word finite before the sequence? Precisely speaking into what I want, What problems would be run into if we had an infinite sequence of axioms?
 A: This isn't actually something we do in general! Both $\mathsf{ZFC}$ and (first-order) $\mathsf{PA}$, two of the most important axiom systems in mathematics, are not finitely axiomatizable. So Schuller is just wrong here. It's been a while but my recollection is that in general he's pretty sloppy about the treatment of foundations - see e.g. this older MSE question (and I recall another question along similar lines, but I can't find it right now). I recommend finding a different source on this topic.
A: I guess the point here is that you want your notion of proof to be effective. Therefore, proofs should be finite objects. Moreover, given an arbitrary finite object, it should be effectively decidable whether it constitutes a proof or not.
The second condition rules out axiom sets which are too complicated. As an extreme example, assume one sets up a calculus and declares that all theorems of first order logic are taken as axioms. Since first-order logic is undecidable, you have no computable way of telling whether an arbitrary formula is an axiom of the system or not. Clearly, this would not be a satisfying situation.
Finite axiom sets are certainly never too complicated in this sense. But there is also nothing wrong with allowing infinite sets of axioms, assuming that the set of axioms is computationally simple. As Noah pointed out, there are well-established first-order theories with infinitely many axioms.
A minimalistic requirement for a notion of proof from infinitely many axioms is then that it is computationally easier to decide what an axioms (or more generally, a proof) is than to decide which formulas are theorems. Otherwise, of what point is a proof? You might as well just list all theorems as axioms.  In the case of PA and ZFC this condition is strongly satisfied: The infinite axiom sets are decidable in linear time (or at a glance, as Doug said in his comment), whereas the resulting set of theorems is undecidable (unless it is inconsistent).
