Seriously, category theory is coming soon. I promise. In the meantime, a friend (who posts on this awesome blog) and I had a conversation about the axiom of choice from a philosophical standpoint. It started when I made the bold claim “whatever, the axiom of choice is obviously true.” Originally, this was meant half as a provocative joke, but really, I think it makes sense. My friend, who I’ll call Steve Neen for the sake of this post, declared, as many mathemematicians do, that he preferred constructive mathematics. After all, how can you say something exists without being able to construct it. This is why mathematicians prefer constructive proofs to non-constructive ones.
I had always sort of accepted this as a reasonable point of view (which I still think it is), but the following thought dawned on me: At their core, each of the axioms of ZFC simply say “there exists a something such that something else.” We may have notation for sets constructed from a specific axiom, but symbology shouldn’t have any bearing on whether or not an axiom is constructive. There is simply nothing intrinsically constructive or non-constructive about any of the axioms.
Of course, Steve then pointed out the difference may not be subjective. Look at what computers can do. They can represent unions, pairs, power sets (though not efficiently), etc. all in an effective manner. But look at the axiom of choice. Computers just aren’t built to do this constructively.
Fair point, Steve. One that had me thinking all of last night. One answer is that what computers can do is somewhat arbitrary, but that feels like a cop out. So here’s my two part answer, that I’m reasonably satisfied with.
- Look at the Axiom of Infinity. It essentially says there is an infinite set. More precisely,
![\exists x[\varnothing\in x\wedge\forall y(y\in x\to y\cup\{y\}\in x)]](http://s0.wp.com/latex.php?latex=%5Cexists+x%5B%5Cvarnothing%5Cin+x%5Cwedge%5Cforall+y%28y%5Cin+x%5Cto+y%5Ccup%5C%7By%5C%7D%5Cin+x%29%5D&bg=ffffff&fg=555555&s=0 "\exists x[\varnothing\in x\wedge\forall y(y\in x\to y\cup\{y\}\in x)]")
- Jokingly, and to prove his point, Steve asked me to pick one element from each coset of
. First of all, it’s important to note that one’s ability to do so is reliant on the axiom of choice. This is one of the reasons I think it should be true. Morally, I can pick out an element from each coset. I told Steve, all I need to do is produce a choice function on the coset. I claimed (again, jokingly) that I had such a function. In fact, I had uncountably many. If he named a coset, I’d consult my function and tell him the answer.
Steve did point out, and is correct in doing so, that just because I claim it exists, and can verify as much as he asks doesn’t mean that I have such a function. But from a computational point of view, it sort of does. Any function that’s consistent with a choice function can be extended. If you accept the axiom of choice, then you can extend it to a choice function. If you don’t, then it can just be extended further, so long as the extension is countable. Regardless, all computations are going to be finite, so from a computational point of view, we never really need the axiom of choice, just as we don’t need the axiom of inifinty. We only need finite choice, which is provable from ZF, even without choice.
. First of all, it’s important to note that one’s ability to do so is reliant on the axiom of choice. This is one of the reasons I think it should be true. Morally, I can pick out an element from each coset. I told Steve, all I need to do is produce a choice function on the coset. I claimed (again, jokingly) that I had such a function. In fact, I had uncountably many. If he named a coset, I’d consult my function and tell him the answer.So that’s a recap of the discussion we had. I’m not entirely convinced anymore that the axiom of choice is “no different” from the other axioms, but it’s an interesting philosophical question I invite you to think about.
