[Math] Difference(s) between an axiom scheme and an axiom

axiomslogic

The basic question which motivated me to write this post is the following,

What is(are) the difference(s) between an axiom scheme and an axiom?

In Margaris's book First Order Mathematical Logic we have the following, enter image description here

However, the difference between an axiom and an axiom scheme is not clearly stated there in the sense that in the book it is not clearly specified exactly what is(are) the property (or properties) that help us to distinguish between an axiom and an axiom scheme.

Question

What is an axiom scheme? Is an axiom scheme different from one of its instances? If so, then in what sense exactly?

Disclaimer

  • I had some discussion with user21820 regarding this question and one can see our discussion here.

  • For a different and more philosophical version of the question see here.

Best Answer

An axiom scheme is simply (but see below) a set - usually infinite, otherwise there's not really much of a point - of axioms. This is different from an axiom, in general, since axioms are sentences in some formal language, and "saying infinitely many things at once" might not be something that language lets us do. Specifically, given a set of sentences $\Sigma$, there may be no single sentence $\varphi$ in our specific formal language which implies every single sentence in $\Sigma$. (Note that we can of course say "Every thing in $\Sigma$ is true" in English, but there's no reason this sequence of words has to have a corresponding sentence in the specific formal language we're working in.)

OK, my first sentence isn't totally honest. Usually we think of an axiom scheme not as just some arbitrary set of sentences, but rather some set of sentences following some pattern. But this isn't relevant to your sub-question of how schemes are different from individual axioms; rather, it just says that some kinds of sets of sentences are nicer than others and deserve special names.

EDIT: Reading your linked posts, it's clear that you're actually interested in my second paragraph above - the one that I didn't elaborate on. So this answer is, I'm sure, very unsatisfying. I don't have time at the moment to elaborate, but I will do so this evening if no one posts a more on-point answer in the meantime. The ultra-short version is: the phrase "of the same form," or "following some pattern," is not formal, and "axiom scheme" is not a formal term; however, it is a useful informal term, and in fact it is easy to formalize (although there is not, in my opinion, any "best" formalization or reason why such should exist).

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