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.
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.