[FOM] n-th order ZFC

Robert Black mongre at gmx.de
Wed Jul 13 06:33:08 EDT 2011

Am 12.07.11 22:18, schrieb Roger Bishop Jones:
> It does not require any metaphysical position at all. Acceptance that 
> "all possible subsets" is meaningful, like acceptance that the concept 
> of natural number is meaningful, is well within the normal standards 
> of mathematics, and is independent of metaphysical ontology. (it is 
> true that philosophers do sometimes refer to the acceptance of the 
> objectivity of truth in some domain as realism, but then some 
> philosophers are unable to accept that one can believe in the 
> objective truth of arithmetic without also believing in the existence 
> of numbers. On that latter point I can assure them as a matter of 
> empirical fact that it is possible).
I don't think you can escape philosophy quite so easily. It's true that 
the view that mathematical sentences have truth values independently of 
whether or not we can discover them (call this 'realism') is distinct 
from the view that the abstract objects mathematical sentences appear to 
talk about really do exist (call this 'platonism'), and neither position 
trivially entails the other. But the fact that you can believe 
arithmetical truths without *thinking that* you are thereby committed to 
the existence of numbers hardly (yet) shows that you can believe 
arithmetical truths without thereby *being* committed to the existence 
of numbers.

As for the normal standards of mathematics, let's take the reals as an 
example. It's a theorem of standard mathematics that there is up to 
isomorphism only one complete ordered field. I think most mathematicians 
unconcerned with foundational issues naively (but in my view also 
correctly) take this theorem to mean just what it says: 'the reals' form 
a completely determinate structure. But there's a different view which 
doesn't of course deny the theorem, but regards it as always relativized 
to a model of first-order set theory, as saying that *for any given 
model of first-order set theory* all the complete ordered fields in that 
model are isomorphic, but that the complete ordered fields in different 
models need not be isomorphic to one another (also true of course). This 
can then be combined with a denial that there is any such thing as the 
intended model of (the first few transfinite ranks of) set theory to 
reach a position where the original result is understood in a much 
weaker way. I should add that there are philosophically and 
mathematically highly skilled people on both sides of this divide: it 
annoys me intensely when people think it's just *obvious* that one side 
is right and the other wrong (not that I'm accusing Roger of that).


Robert Black

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