[FOM] Falsify Platonism?

[Daniel Méhkeri]

Timothy Chow says:

> All right, having said all that, why don't I think that an inconsistency 
> in PA would "falsify platonism"?  Well, the way I've laid things out 
> here, it's easy to see where the loopholes are.  The one that stands 
> out to me is the assumption that all first-order sentences of 
> arithmetic coherently express legitimate properties of the natural
> numbers.  This assumption sure seems obvious, but if we had an 
> inconsistency in PA staring us in the fact, then I think it would 
> seem less obvious.  Supposing that the inconsistency could be avoided 
> by dropping down to a weaker induction axiom, that would be a tempting 
> route to take.  Then we could still claim that the Peano axioms 
> define the natural numbers, and that induction applies to "all 
> properties," but that some of the formulas of first-order arithmetic 
> do not, despite appearances, legitimately express actual "properties" 
> of the natural numbers. This route is uncomfortable, perhaps, but 
> mathematicians could probably learn to live with it.  

All of this seems true, except for the implicit claim that what is left is 
"Platonism" in any sense. It's not even constructivism. To deny meaning to 
quantifiers over all of N is to adopt some form of finitism, if not 
ultrafinitism. No? 

In fact, what's the next conceptual step down from epsilon_0? omega^omega. 
So, the plausible weaker induction axiom would be Sigma_1 (if not lower), 
and this is exactly finitism in Tait's sense. Think about Goodstein's 
primitive recursive arithmetic. This has induction for "all properties",
as you say. It's just that there are no quantifiers. (Or any other 
logical connectives for that matter: they are definable.) This seems like 
the appropriate fallback, because wouldn't it be felt that "logic" 
as we know it had been shown to be nonsense?

> After all, this is approximately the same approach that 
> mathematicians have taken with the set-theoretic antinomies.

This analogy was made by others, but I'm not sure. Set theory was very
young and not established at the time. Also that was long ago and those
mathematicians are all dead. Wouldn't an antinomy in even ZFC+Mahlo
have a completely different effect now?

Regards,
Daniel



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