[FOM] "Proof" of the consistency of PA published by Oxford UP

Timothy Y. Chow tchow at alum.mit.edu
Fri Mar 6 18:11:45 EST 2015

Richard Heck wrote:

> If there's something interesting here, it's the way the semantics he 
> develops doesn't require there to be a single infinite model, but only a 
> succession of every-larger finite models. There are antecedents to that 
> sort of idea in modal structuralist views, I believe, of the sort 
> developed by Hellman, and perhaps more than antecedents. Maybe there are 
> more developed forms of this idea, too, and if so I'd be interested to 
> know where.

There is of course a long tradition in philosophy of distinguishing 
between "potential infinity" and "actual infinity."  In modern 
mathematics, this distinction doesn't seem to exist.  The closest thing to 
"potential infinity" seems to be an axiomatic system that lacks anything 
that could be identified as an explicit "axiom of infinity," yet admits 
only (actually) infinite models.  (PA would be an example.)  But for 
example, I've never seen anyone define two separate axioms and declare one 
of them to be an "axiom of potential infinity" and the other an "axiom of 
actual infinity."

I'm wondering if anyone can come up with (or has already come up with) 
candidates for two such axioms, the former demonstrably weaker than the 
latter, such that the consistency of PA can be proved using only the 
"axiom of potential infinity."  Doing this might be pleasing to those who 
not only perceive an important distinction between potential and actual 
infinity but go so far as to reject the latter while accepting the former. 
(McCall seems to be one such person, if I read him correctly.)


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