FOM: consistency equals existence

[Robert Black]

I am grateful to Vladimir Sazonov for his comments on my post of 18
February. However, I must point out that I was there exploring the
implications of positions which I would not necessarily want to adopt. In
fact I am tempted by a realism/platonism so extreme that it would make
Professor Sazonov's hair stand on end.

Commenting on this realist view, Steve writes:

>However, it seems to me that this idea founders on the Continuum
>Hypothesis.  Recall that there are sentences A and B of second order
>logic which are satisfiable (i.e., satisfiable in the standard or
>full-power-set semantics) if and only if 2^aleph_0 = aleph_1 and
>2^aleph_0 > aleph_1, respectively.  Now we know from G"odel and Cohen
>that both A and B ``could be true'', but only one of them is
>satisfiable in ``the'' set-theoretic universe.  It seems to me that
>this is a severe problem for Black's idea above.

But my answer to this is given by Steve's own immediately following sentence:

>Of course Cohen's
>work deals ``only'' with first-order set theory

On the realist view, since second-order ZF is quasicategorical, CH is
either true in all (full) models of it or false in all (full) models. If
it's true then it couldn't have been false, and if it's false then it
couldn't have been true. There are of course structures satisfying
first-order ZFC+CH and structures satisfying first-order ZFC+not-CH, but
either none of the former or none of the latter are also models for
second-order ZF, because sets which 'ought' to have been there have been
left out. (As Volker Halbach remarked to me in an off-FOM mail there are
also models of first-order Peano Arithmetic satisfying not-Con(PA), but I
take it Steve wouldn't want to conclude from this that PA could have been
inconsistent!.)

I didn't of course mean to imply that anything like the full strength of
ZFC is needed to prove the completeness theorem, and it is interesting to
learn that exactly (in a certain sense) what is required is WKL_0. (I note
this is proved on p.140 of Steve's book, and confess to not having got that
far.) But I think my point still stands; if we're to use the completeness
theorem to argue for a version of 'consistency equals existence' we need
not merely the fact that the theorem is provable in a formal system (here
WKL_0) but also that that system has a model (in the case of WKL_0
presumably a model whose objects can be identified with natural numbers and
sets thereof).

Robert

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845