Actual and Potential infinite

JOSEPH SHIPMAN joeshipman at
Sun Feb 5 10:32:39 EST 2023

Most of the discussion that is going on about “actual” and “potential” infinite does not seem to involve any theorems, so let me ask a couple of questions that focus on mathematical practice.

1) Is there any proposition statable in the language of first order arithmetic, which is a theorem of ZF but not of PA, which can be said not to require “actual infinity” to prove?
2) Can you name two theorems T1 and T2 statable in the language of analysis (second order arithmetic), whose *statements* are of the same logical type, such that T1 requires “potential infinity“ but not “actual infinity”, but T2 requires “actual infinity”, for some reasonable meaning of “requires”?

I am struggling to understand how this discussion, when reduced to questions of what can be proven, is about anything other than the cut point in the hierarchy of logical strength that appears between PA and PA+Con(PA).

— JS

Sent from my iPhone

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