FOM: CH and 2nd-order validity

Robert Black Robert.Black at
Wed Oct 18 18:38:03 EDT 2000

>Professor Black,
>You said:
>``It *is* of course provable in ZFC plus an inaccessible, and I don't think
>it's too unreasonable to say that that proof formalizes the informal
>notions that provide the principal basis for our confidence in the
>consistency of ZFC. So that would be an example of a rigorous proof which
>is not a proof in ZFC. ''
>Is it not the case that Professor Kanovei may object that the proof that is
>carried out is a proof that Inacc --> Cons(ZFC) ?  (I take here Inacc to be
>the formula
>that formalizes the large cardinal axiom ``there is an inaccessible
>cardinal''.)  May he
>not reasonably assert that such a proof is not a proof of Cons(ZFC)?
>Matt Insall

I think there's quite a difference here between using really large
cardinals  (say measurable and upwards) and just using one inaccessible.
The large large cardinal axioms might well be regarded as extravagantly
interesting hypotheses which may or may not be consistent, let alone true,
whereas the existence of at least one inaccessible looks very safe, and
very much garanteed by the sort of intuitions which we trust for the
consistency of ZFC itself.

In ZFC one can prove the consistency of Zermelo set theory (i.e. with just
separation instead of replacement). If this is a legitimate mathematical
proof of the result (rather than just a proof in ZC of the conditional if
replacement then cons(ZC)), I can't see why one should jib at proving the
consistency of ZFC in ZFC+Inacc. I just don't see anything magical about
ZFC as opposed to ZC (much weaker, but still enough for all ordinary
mathematics) and ZFC+Inacc (stronger, but onceptually only just). Contrast:
Scott's theorem tells us that if there's a measurable cardinal then V=L is
false. Someone might well insist that this is just a proof of the
conditional, and not an unconditional proof that not every set is
constructible - and a fortiori with the much stronger hypotheses that set
theorists play with these days.

However, should a set theorist (John Steel?) disagree with this, I would defer.

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

tel. 0115-951 5845

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