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 is not quite such an 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 infintiely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s system NF was the only significant system of set theory that was not known to be equiconsistent with ZFC,
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
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) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s NF was the only significant system of set theory for which this had not been done. I don’t know if progress has been made on that since.
(ETA: I found this review of NF from 2011. Its consistency was still open then.)
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Randall Holmes says he has a proof of the consistency of NF relative to ZFC (and in fact something weaker, I think). He’s said this for a while, he’s published a few versions of his proof (mostly different in presentation in the interests of clarity, rather than patching bugs), and I think the general feeling is that he probably does have a proof but it hasn’t yet been thoroughly checked by others. (Who may be holding off because he’s still changing his mind about the best way of writing it down.)
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.
Yes, ZFC is not quite such an 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 infintiely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s system NF was the only significant system of set theory that was not known to be equiconsistent with ZFC,
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
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) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s NF was the only significant system of set theory for which this had not been done. I don’t know if progress has been made on that since.
(ETA: I found this review of NF from 2011. Its consistency was still open then.)
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Randall Holmes says he has a proof of the consistency of NF relative to ZFC (and in fact something weaker, I think). He’s said this for a while, he’s published a few versions of his proof (mostly different in presentation in the interests of clarity, rather than patching bugs), and I think the general feeling is that he probably does have a proof but it hasn’t yet been thoroughly checked by others. (Who may be holding off because he’s still changing his mind about the best way of writing it down.)