[FOM] Potential and Actual Infinity

[Joseph Shipman]

It's not clear how to make sense of this question. I can prove Con(PA) from the axiom that epsilon_0 is well-ordered, or from Goodstein's Theorem, or from various pi^0_1 statements which do not themselves speak about or seem to entail an actually infinite set. However, if we want to find a proof of Con(PA) from within our official formal foundational system ZFC, it is absolutely necessary that somewhere in that proof, the Axiom of Infinity is used, and that states that an actually infinite set exists. So there is a strong technical sense in which the answer to your question is "Not if ZFC-derivable is the standard for provability".

-- JS

Sent from my iPhone

> On Mar 13, 2015, at 1:55 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
> 
> Joseph Shipman wrote:
> 
>> This discussion seems to be making too much of a simple point. In Peano Arithmetic, which can be equivalently formalized by taking ZF and replacing the axiom of Infinity with its negation, there is no actual infinity.
> 
> Yes, this is what I said in my initial post, and I think I was the first one to bring up the term "potential infinity" in the current discussion.
> 
> The question I raised in that same post, however, does not seem to have been answered yet.  Namely, is there a way to prove the consistency of PA assuming only "potential infinity"?  What we might call the classical approach to potential infinity turns this question into, can PA prove its own consistency?  So the classical form of the question has a negative answer.  However, McCall seems to want to claim some kind of positive answer.  Arnon Avron also claimed that potential infinity was all that was needed to prove the consistency of PA, but unless I missed something, has not responded to my request for a more formal justification of this claim.
> 
> Tim
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