Talk:Axiom of countability

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I'd like to add that in many cases, the interesting mathematical objects are those that satisfy certain countability axioms. I don't really know how to make that statement precise, though.

Prumpf 19:56, 16 Aug 2004 (UTC)

Yeah, it's kinda hard to state except maybe through a good set of examples and counterexamples. — Fropuff 20:14, 2004 Aug 16 (UTC)

Why are they "not axioms in today's sense of the word"? Most certainly they are not axioms in the sense in which that word has long (2500 years?) been used in philosophy, but in modern mathematics and axiom is really just a statement of a property that some objects possess and others don't. Michael Hardy 22:17, 18 Aug 2004 (UTC)

Hmm. I'm not sure of that last statement. I believe that for most (non-meta)mathematicians, the ZFC axioms are still axioms in the classical sense, even though there's a vocal minority that likes to exclude AC. However, I think the other use of axioms is that several axioms (not just one, since that would be just a property) are used to define certain classes for consideration. The real numbers are defined by their axiomatisation, up to isomorphism, and so are the topological spaces. However, "axioms" of countability define no such structure, and they're not statements so obvious they could be accepted as self-evident either.
In essence, I think this is a linguistic issue, but I think if someone came up with a new "axiom of countability" today they'd call it a countability property rather than an axiom. In fact, I can think of very few recent papers I've read that introduce new axioms at all. I would go as far as claiming that "axiom" is used mostly for historical axiomatisations these days, much as the principle of complete induction now is more properly called a theorem about the natural numbers. Universal properties really are axiomatisations of certain isomorphism classes, but they're not called that.
(There's a notable exception in certain uniqueness proofs, such as the proof that homology theories can be uniquely axiomatised (Igusa's axiomatisation of higher Franz-Reidemeister torsion is a very recent example of introducing axioms in this sense, I guess) Sorry this got a bit long. It would feel wrong now, to me, to call a countability/separability property axiom, and thus I don't think they're axioms in today's sense of the word. I'll try to reword it though.
Prumpf 23:19, 18 Aug 2004 (UTC)
Perhaps you're right about ZFC, but people routinely speak of the "axioms" of group theory and the like. They don't mean they're "self-evident" statments about groups, but only that they are the criteria according to which one sees what is a "group" and what is not. Michael Hardy 02:40, 19 Aug 2004 (UTC)

Mood

I totally agree with Prumpf's “However, "axioms" of countability define no such structure, and they're not statements so obvious they could be accepted as self-evident either.” argument. And I believe we should re-write this article in the following mood:

Axiom of countability is an axiom to define second-countable space:
1. (Axiom of union of any)The union of any collection of sets in $\tau$ is also in $\tau$.
2. (Axiom of intersection of finite)The intersection of any 2 sets in $\tau$ is also in $\tau$.
3. (Axiom of countability-2nd)There exist a countable collection of $\tau$, such that every members of $\tau$ can be written as a union of elements of this collection.

By such kinds of definition the "axiom of countability" does define 2 structure - first and second-countable space, and naturally, become axioms. --––虞海 (Yú Hǎi) 11:08, 15 October 2010 (UTC)