[FOM] Potential and Actual Infinity

Timothy Y. Chow tchow at alum.mit.edu
Fri Mar 13 13:55:43 EDT 2015

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.


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