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?
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.
If we have an infinite sequence of axioms, then we need some sort of method to define how the sequence of axioms could get produced given an &quot;infinite amount&quot; of time.  Of course though, there is no such thing as an infinite amount of time that anyone has ever known from experience at least.  I read also one person analyzing Galielo a while back also saying that he never used the term &quot;infinite number&quot;, and regarded that phrase as a contradiction ( the concept of &quot;infinite number&quot; probably could get contested at least somewhat successfully by an ultrafinist argument).  Also, no one has ever been, nor will be ever able to, produce an infinite sequence.  And again, we need some method to define all of those axioms.
A finite sequence of axioms doesn't have these (potential) issues.  At least for some small finite sequences, we can list the entire sequence, and thus determine at a glance whether or not some well-formed formula belongs to the sequence of axioms.  We don't need some method to define it.  There is no need, ever, to get into addressing perhaps a very well-founded objection of there not existing an &quot;infinite amount&quot; or &quot;infinite number&quot;, since those concepts are irrelevant to what any given axiomatic system consists of.  Finite sequences of well-formed formulas exist.  Infinite sequences of well-formed formulas existing is at least potentially false.  Therefore, using a finite sequence of axioms has at least a few advantages, and there can exist strong reasons to prefer them.
Additionally, a finite sequence of axioms can get shown to have an independent set of axioms by a finite process.  At the very least, the process of showing such seems potentially more complicated for an infinite set of axioms.
A computer can process some or many finite sequences of axioms.  A computer can't process an infinite sequence of axioms, since there is no computer that can perform an &quot;infinite number&quot; of computations and will only for a finite amount of time, correct?