I'm sorry if the title sounds overly confrontational. There will be some confusion and a bit of anger throughout the post, but I hope it will be clear what I'm asking by the end of it.
I've been reading about the axiom of choice again. And then about the axiom of determinancy. And then about the axiom of infinity. I'd done it before, but this time it's really gotten into me. What am I doing? Why am I studying all of this? Is my study and potential future work needed? Why should anyone want to pay me for it?
Obviously, a lot of mathematicians' work is needed, useful and worthy of appreciation. But who needs the existence and uniqueness of algebraic closures? Someone might say that they are beautiful in their simplicity. Are they? Well, maybe, but is the Banach-Tarski decomposition beautiful? It's total bullcrap, and everyone knows it. It's mischief, trickery. And we have to put up with this if we want the beauty and simplicity of algebraic closures that always exist and are unique for our convenience.
But convenience in doing what? Isn't it all just made up? What does the taxpayer get in return from paying for the existence and uniqueness of algebraic closures? She certainly won't use them directly. Is there really anything in it for her at all? What if she's a physicist? I know little about physics, but can't imagine how a physicist could need that. It seems to me that anything that has something to do with choice is doomed to be useless in describing the real world. It seems to me that we are proving all these theorems just to please ourselves. Am I wrong?
And do we even know that the algebraic closures can exist and be unique? No, we're playing this game without even knowing that the rules aren't inconsistent.
Apparently, I've read, most or all of the concrete mathematics needed to describe the world needs only the potential infinity. I understand that without the axiom of infinity the definable infinite sets can be spoken of as formulas. Apparently, we don't need to call them sets. Just like we don't need to call the class of ordinal numbers to be a set to speak about it in ZFC. Why on earth wouldn't we do that? To have Goodstein's theorem? Come on.
Why not get rid of all these junk objects we don't really need for anything? The integers can be constructed from finite sets, from the we can construct the rationals, and then the reals and so on. But now we have the constraint of definability to save us from a lot of junk. Wouldn't it be good?
I was recruited by people who told me there existed unboundedly many infinities. Now I just feel tricked and used.
The above is a series of not entirely coherent accusations and questions that may seem rhetorical or stupid, I know. But they're not, at least not rhetorical. I'm worried that what I'm doing might be useless. I'm not going to stop studying mathematics because it's too late. I'm 25 years old and I can't do anything else. But I would like to feel good about my education.