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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?

MJD
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tryst with freedom
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2 Answers2

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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.

Noah Schweber
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  • This doesn't answer the question at all and instead engages in a reputation attack against the original source. Possibly worse, it doesn't even consider that there could be advantages to consider a finite sequence of axioms, so the question gets suggested as not worth the time answering whatsoever. Also, I don't know if Schuller were defining what an axiom system were for all acknowledged as logicians, or just for his lecture series, do you? That is, what is the intended scope of Schuller's definition? – Doug Spoonwood May 06 '21 at 18:36
  • The point of the lecture series was to explain the 'geometry' of physics through some what deep mathematical framework. And, yeah you are right on the last point, schuller did say that it would be much more complicated if we were to be more formal (eg: indexing the axiom set use numbers for instance). @Doug – tryst with freedom May 06 '21 at 18:41
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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).

Timo
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