FOM: CH and 2nd-order validity

[Robert Black]

Reply to Vladimir Kanovei:

Of course (with qualifications about highly confirmed conjectures which
don't matter here) we *establish* mathematical truth by proof, so I can't
give you examples of *accepted* mathematical truths lacking proof. But it
doesn't follow from that that there aren't any mathematical truths lacking
proof, just that we don't accept them until we've found a proof (and so if
they don't have a proof we'll never accept them). You might just as well
argue that because I can't give you a properly verified example of any
truth which I do not know to be such it follows that I'm omniscient.

Cons(ZFC) is a statement of arithmetic, and you say you believe it (or that
one should believe it, and I'm sure you do what you should), which means
you take it to be true, i.e. a fact. What kind of fact it could be other
than a mathematical fact is beyond me.

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. (Of course Harvey's recent work provides lots of
further examples, but I'm not quoting them because someone might deny that
something proved with the aid of large cardinal hypotheses has really been
proved.)

Robert




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

tel. 0115-951 5845