[FOM] FOM Digest, Vol 205, Issue 50: Tim Chow on Bivalence and Unknowability

[Timothy Y. Chow]

Alan Weir wrote:

> Yes that seems right but I think my point still stands: not caring about 
> any question specifically about a particular domain seems no reason to 
> reject bivalence, and unknowability a good reason only if you have some 
> sort of anti-realist stance for that particular the domain in question.

Yes, I agree.  I would only add that I think that anti-realist stances are 
often driven by a feeling that the entire domain is devoid of interest and 
is dispensable.

> The important thing, of course, is whether such a reason for 
> differentiating arithmetic from infinitistic set theory can be made 
> clearer and be coherent.  I, for one, am sceptical.

There is already a clear technical definition of what an arithmetic 
statement is, that does not depend on infinite set theory.  So one option 
is simply to dig in one's heels and insist that arithmetic statements are 
meaningful and statements in set theory are not, unless they can be 
translated into "equivalent" arithmetical statements.  This would be 
clear, but whether it would be "coherent" depends on what your standards 
for "coherence" are.  Is it enough for such a person to shift the burden 
of proof away from themselves (to articulate a reason for rejecting 
infinite set theory) to the proponent of infinite set theory (to 
articulate a reason for accepting infinite set theory)?  By historical 
standards infinite set theory is after all a Johnny-come-lately, and it is 
not hard to find anti-infinitary statements by mathematical luminaries, 
especially from past generations.

Tim