[FOM] Sharp mathematical distinction between potential and actual infinity

JoeShipman@aol.com JoeShipman at aol.com
Sun Sep 28 19:44:44 EDT 2003

In mathematical practice, nothing hinges on whether an infinity is "potential" or "actual".  What actually matters is whether an Axiom of Infinity is used or not.  For definiteness, let us restrict our attention to the 99% of mathematical practice formalizable in ZFC.

Then we have:
1) Any theorem of ZFC which has a ZFC-proof that does not use the Axiom of Infinity does not require an "actual infinity".

2) Any theorem of ZFC, every proof of which uses the Axiom of Infinity (for example, the theorem "Con(PA)"), may be said to require "actual infinity".  

3) I do not see a clear meaning to "potential infinity" essentially different from "quantifier ranging over natural numbers without further restriction".  The term "potential infinity" may be said to be applicable to results of the form "for any integer n, there exists an integer m > n such that Phi(m)", where Phi is a one-place predicate.  Results provable in certain restricted subsystems of arithmetic may be said to "not require potential infinity", but I would rather let someone else suggest a precise formulation.  

Linguistically, one can point to the distinction between "for ANY n, there exists m>n Phi(m)" and "for ALL n, there exists m>n Phi(m)" as illustrating potential and actual infinity, respectively, and make a corresponding distinction between the function taking n to m (potential) and the graph of the function (actual), but this is NOT an important distinction in mathematical practice, where the universal quantifier symbol is indifferently pronounced "for all" and "for any", with most mathematicians unperturbed about the usage.

-- Joseph Shipman

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