FOM: trivial amalgamation; Hilbertism; consistency equals existence; applications to analysis; practical completeness
neilt at mercutio.cohums.ohio-state.edu
Thu Feb 17 10:10:44 EST 2000
Steve Simpson wrote:
> There is an old dictum (Hilbert's?) that existence (of mathematical
> objects) is the same as consistency.
> [ Here is the argument. Let S be a system of mathematical objects
> whose existence is in question. Let T be a set of axioms describing
> all of the properties that S is to have. Since mathematical objects
> are completely described and axiomatized by their properties, T ``is''
> S. But according to the G"odel Completeness Theorem, T is consistent
> if and only if S ``exists'', in the sense that a model of T exists.
> So, existence (of S) equals consistency (of T). ]
> Now, this dictum has a lot of appeal. In fact, it is a rigorous
> theorem, the G"odel Completeness Theorem. Why does Steel apparently
> think it is ``strange''?
I don't know what Steel's reasons would be, but here are mine. The G"odel
Completeness Theorem guarantees only that if T is consistent, then some
*countable* model of T exists. So if one's intended model S is
uncountable, then the consistency of one's theory that describes S does
*not* provide any guarantee that S itself exists. Moreover, even if one
can ascribe a cardinality k in advance to one's intended model S,
appealing to the upward L"owenheim-Skolem-Tarski theorem (which implies
that *some* model of cardinality k exists for one's theory T) won't turn
the trick, unless one can also show T to be k-categorical. So T might say
all the right things without being about the right things.
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