FOM: consistency equals existence

Stephen G Simpson simpson at
Thu Feb 17 11:27:31 EST 2000

Neil Tennant writes:

 > The G"odel Completeness Theorem guarantees only that if T is
 > consistent, then some *countable* model of T exists. 

Why ``only''?  All that I needed for my consistency-equals-existence
argument was that *some* model of T exists, provided T is consistent.

The premise of my argument was that S is completely axiomatized by its
properties as specified by T.  This does not mean that T is a complete
theory, or that T is in any sense categorical.  It only means that S
is axiomatized by T.  For instance, suppose S is a certain kind of
projective plane with certain combinatorial properties, whose
existence is in question.  Then we can axiomatize S in terms of
appropriate primitives -- points, lines, etc -- and then S exists if
and only if there exists a model of the axioms.  This can be *any*
model of the axioms, even a countable one, not necessarily the
``intended'' model, whatever that means.  As Hilbert said in a famous
pronouncement, the points can be tables and the lines can be beer
mugs, for all we care.

 > 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.

But wait.  Let's play by the rules.  The rules say that S was to be
specified by its axioms.  If uncountability was one of the properties
that we intended for S, then shouldn't we have included uncountability
among the axioms?  And in this case, if the axioms are consistent,
then any instance of S that we get from any model of the axioms *will*
be uncountable, in an appropriate sense, namely, in the sense of the
model, even if the model is countable when viewed from outside the

This reminds me of the ``second order logic is a myth'' discussion
that took place on FOM last year.  My point now is as it was then.
Namely, predicate calculus and the G"odel Completeness Theorem need to
be taken seriously, even when applied to concepts such as
uncountability that are normally defined in a context of second order
logic or set theory.  An instance of this is the Skolem Paradox.

 > 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.

Same rejoinder as above.  Categoricity doesn't matter.  All we want is
*some* model of T, *any* model of T, in order for a system of objects
S to exist which has *all* the properties that we requested in our
axioms for S.

I know that this a fairly radical anti-set-theoretic-realism stance.
And it is not necessarily my own stance.  I present it as an antidote
to Steel's rhetorical dismissal of ``Hilbertism'' etc.

This is no criticism of Steel.  I know very well that Steel was
transcribing ``The Speech'' and not necessarily trying to give a
balanced presentation of competing philosophies of mathematics.

-- Steve

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