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In the definition of a vector space, one often uses the word axiom to say that anything that satisfies the given axioms is a vector space. But the word axiom usually refers to a statement that is believed to be true without further justification. So, checking that addition and scalar multiplication satisfy the list doesn't seem to be necessary. In that sense, it doesn't make much sense to say the sentence "anything that satisfies these axioms".

The same thing happens when talking about other mathematical structures. For example, take the definition of a group. It does't make much sense to say that having an identity, inverse and an associative operation are axioms. I think it makes more sense to use the word property. Saying that anything that satisfies the given properties is a group or a vector space. It also makes sense to check the properties to see that they satisfy it.

Am I missing something here or can the use of the word axiom be replaced by the word property?

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    $\begingroup$ Axiom: Yes, you can. $\endgroup$
    – plop
    Commented Sep 1, 2022 at 22:24
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    $\begingroup$ If you enjoy thinking about this, I would recommend learning model theory. Formally, "group" and "model of ZFC" are the same type of thing: a model of the "group axioms" and "axioms of ZFC" respectively. (This video by Thomas Kern is not a bad starting point.) $\endgroup$ Commented Sep 2, 2022 at 0:11
  • $\begingroup$ @AkivaWeinberger Thanks a lot! I will check the video out and start reading up on model theory. $\endgroup$
    – Seeker
    Commented Sep 2, 2022 at 0:30

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To some extent, this is a matter of opinion: but here is my opinion, which I hope at least some mathematicians and logicians will find not too objectionable (it is a formalist take on the problem, with a bit of a twist at the end; apologies to mathematical historians for my fairly simplistic historical statements).

In modern mathematics, we don't use the term axiom in the quite same way as the ancient Greeks used it or as many modern philosophers do. The ancient Greeks thought of as their theory of geometry as a description of an ideal model of the geometry of space we deal with in the physical world. Moreover, they (or at least many of them) thought of that ideal model as actually existing in some way. Hence, they could view the axioms of geometry as postulates about that extant model that need no further justification. Modern philosophers dealing with the real world or aspects thereof naturally and understandably take the same approach, because they want to talk about a given extant world. In mathematics this approach is often called Platonism and it is not so clear that it is appropriate.

In modern mathematics, we are aware of examples like non-Euclidean geometries and set-theoretic issues like the axiom of choice, which make (at least some of) us less convinced that there is some one fixed ideal mathematical universe that we want to work with. So, in modern mathematics, the term "axiom" is usually used to refer to a property of a class of mathematical structures that serves as part of a useful theory that can be applied to interesting problems. Hence the axioms of group theory represent a set of assumptions about a binary operator on a set that has been found to be widely applicable and very useful.

The twist is that you can reconcile the modern axiomatic method with the Platonistic view by believing that, in some sense, there is an ideal world of groups out there. The axioms of group theory then just become the reflections in Plato's cave of that extant world containing all groups. Likewise for any other class of structures, like sets or vector spaces or what have you. I personally disbelieve the existence of that ideal world from a philosophical point of view, but that doesn't make any difference in mathematical practice: if you take the Platonist view and I take the formalist view, our notions of mathematical truth coincide, so we can work together as mathematicians, even though our philosophical views are quite different.

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Tl;dr: axioms are properties, but not every property is an axiom(of a given theory).

In mathematics, when we speak of "axioms", we typically mean the defining properties of some concept.

For example, associativity of addition is not just a property of a vector space: it is a necessary condition for something to be a vector space in the first place; if you have a structure whose "addition" is not associative, this structure is not a vector space. In contrast, being two-dimensional or having more than 17 elements are properties that some vector spaces have, while others don't.

This means that if you give me a vector space, I can take for granted that the addition is associative (without justification, as you say), but not that the vector space has more than 17 elements.


More specifically in logic, by axioms of a theory we usually mean a (hopefully somewhat concise) family of sentences from which all of the theory follows in a given deductive system. For example, we have the Zermelo-Fraenkel axioms of set theory, but your vector space example also fits into this framework; there are also non-first order axiomatisations, such as the axioms of a complete ordered field.

In this context, it is not, strictly speaking, rigorous to call a given sentence an axiom of a theory, since a given theory can have many (even disjoint) axiomatisations, and any consequence of this theory can be taken as an axiom (every theory is axiomatised by itself in its entirety, after all). For example, if you give me a vector space, I can also take for granted that I have the exchange property for linear independence, which is not usually taken as an axiom, because I know that it is a theorem that follows from the axioms of a vector space.

Still, in practice, the axiom system is often implicitly fixed (sometimes up to minor variation which does not really matter), so we still say that something is "an axiom", meaning that it is one of the basic properties which we "take for granted", as opposed to a theorem.

(Sometimes there are multiple fundamentally different reasonable, equivalent axiomatisations. In this case, of course, we use adjectives.)

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