FOM: Con(ZFC), simplicity, intuitiveness, etc.

Neil Tennant neilt at
Tue Aug 25 19:54:28 EDT 1998

In my first posting on this topic, I made the following points:

1) There is a "synthetic" claim, which might be called Con(ZFC), which
is about arrays of symbolic objects.

2) There is another version of the claim, the version that is usually
denoted by "Con(ZFC)", which involves quantification over [G"odel]
numbers [for those symbolic objects].

3) In neither sense (1) nor sense (2) does our grasp of
Con(ZFC)---that is, our understanding of its meaning---require or
presuppose any understanding of sets or of set theory.

As far as I can tell, none of Steve's rejoinders has taken issue with
any of these points.

I made the following further point:

4) In order to understand Con(ZFC) in sense (2) above as making the
claim that ZFC is consistent, one has to know at least something about
numeralwise representability of recursive predicates.

Steve's response to (4) was to advert to (1). But I had already
acknowledged (1), and had focused on (2) when making point (4).

Let me try to strengthen the case for (4), by interpolating a
precursor claim:

(4*) In order to prove that Con(ZFC) is independent of ZFC, one has to
take Con(ZFC) in version (2) above (the version involving
quantification over G"odel numbers).

Both (4*) and (4) are important in the context of the discussion of
the simplicity and intuitiveness of independent statements. For, we are
looking for simple, intuitive statements S and very strong theories T
such that S is independent of T. One precondition for simplicity and
intuitiveness of S is that one should not have to know any deep results in
order simply to understand what S says. (This is what the "corridor
test" is all about.)

Steve ought to see my points above as actually strengthening the case
for the view that Harvey's recent independence results represent an
essential advance in the state of the art. The simple and intuitive
reading that the independent sentence S sustains when first presented
should be all that is required for the proof of independence itself.
This is not the case, however, with Con(ZFC) and its ilk. The simple
and intuitive reading of Con(ZFC), should it be written out in
primitive notation when first presented, is that of some longwinded
negative existential claim in first-order arithmetic. But then in
order to get independence of Con(ZFC), we would have to make a kind of
Gestalt switch in our understanding of it, via the G"odel coding and
representability. Indeed, hardly anyone ever encounters Con(ZFC)
except as composed via the representing predicates! It is as though
one needs to pinch oneself to remind oneself that Con(ZFC) is,
strictly speaking, just a long and dull claim about natural numbers.

The only way to make Con(ZFC) (in version (2)) display any degree of
the sort of conceptual simplicity prized by Steve is to know some
non-trivial facts about numeralwise representability.  I think that
point now sticks, given appropriate emphasis on *why* it is version
(2) that one has to deal with.

I trust that this last reply of mine to Steve on this matter will not
have tried Martin's patience! I feel, as I'm sure Steve does, that
these philosophical disputes are of some substance.
Neil Tennant


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