It is (I think) true that if you try to come up with an alternative foundation for mathematics you are likely to get something that's equivalent to some subset of ZFC perhaps augmented with some kind of large cardinal axiom. But that doesn't mean that ZFC is inevitable, it means that if you construct two theories both intended to "support" all of mathematics without too much extravagance, you can often more or less implement one inside the other.
But that doesn't mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes's book "Elementary set theory with a universal set") which I'll call NFUIC henceforward. This is consistent relative to ZFC, and indeed relative to something rather weaker than ZFC, and NFUIC + "all Cantorian sets are strongly Cantorian" (never mind exactly what that means) is equiconsistent with ZFC + some reasonably plausible large-cardinal axioms. OK, fine, so there's a sense in which NFUIC is ZFC-like, at least as regards consistency strength. But NFUIC's sets are most definitely not the same as ZFC's sets. NFUIC has a universal set and ZFC doesn't; ZFC's sets are the same sizes as their sets-of-singletons and NFUIC's often aren't; NFU has lots and lots of urelements and ZFC has just the single empty set; etc. NFUIC is very unlike ZFC despite these relationships in terms of consistency strength.
[EDITED to add:] Here's an analogy. You get the same computable functions whether you start with (1) Turing machines, (2) register machines, (3) lambda calculus, or (4) Post production systems. But those are still four very different foundations for computing, they suggest quite different possible hardware realizations and different kinds of notation, they have quite different performance characteristics, etc. (The execution times are admittedly all bounded by polynomials in one another. We could add (5) quantum Turing machines, in which case that would no longer be known to be true.) It's very interesting that these all turn out to be equivalent in power in some sense, but I wouldn't call that convergence or suggest that it tells us that (e.g.) lambda-calculus terms have any sort of more exalted metaphysical status than they would if it weren't for that equivalence.
But that doesn't mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes's book "Elementary set theory with a universal set") which I'll call NFUIC henceforward.
Yes, ZFC may be not quite such a starkly isolated landmark of thinginess as computability is, which is why I said "a strong tendency". And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infinitely deep) c...
Among my friends interested in rationality, effective altruism, and existential risk reduction, I often hear: "If you want to have a real positive impact on the world, grad school is a waste of time. It's better to use deliberate practice to learn whatever you need instead of working within the confines of an institution."
While I'd agree that grad school will not make you do good for the world, if you're a self-driven person who can spend time in a PhD program deliberately acquiring skills and connections for making a positive difference, I think you can make grad school a highly productive path, perhaps more so than many alternatives. In this post, I want to share some advice that I've been repeating a lot lately for how to do this:
That's all I have for now. The main sentiment behind most of this, I think, is that you have to be deliberate to get the most out of a PhD program, rather than passively expecting it to make you into anything in particular. Grad school still isn't for everyone, and far from it. But if you were seriously considering it at some point, and "do something more useful" felt like a compelling reason not to go, be sure to first consider the most useful version of grad that you could reliably make for yourself... and then decide whether or not to do it.
Please email me (lastname@thisdomain.com) if you have more ideas for getting the most out of grad school!