Actual and Potential infinite

[Matthias]

Without a doubt, there are several answers to your questions. I think 
that this is not a question of truth and provability, but it's about the 
"nature of mathematical reality" (as Marcin Mostowski said in an old 
post to this topic). It is basically a model theoretic question and the 
constraints are given by the possibility to interpret expressions in 
such a model. So the task is to give a model that has no infinite sets, 
only finite sets which can be extended if required. My aim is to devolp 
such models and to show that almost all of mathematics can be 
interpreted in them.

Consider first-order logic: A Tarskian model starts with a carrier set. 
The usual assumption is that such sets exist, either in a universe of 
sets, or they exist nively. The potential infinite requires another 
starting point such as direct or inverse systems. These systems can be 
understood dynamically as growing. The challenge is to give an 
interpretation that uses only finite sets in these systems for all kind 
of mathematical entities such as numbers, sets, relations, functions, 
function spaces and ranges for quantification. In particular, the 
universal quantifier cannot be interpreted naively by "all natural 
numbers", but has to refer to a sufficiently large finite set.

In my intital post about the potential infinite I wrote a little more 
about it, so I don't repeat it here.

Kind regards,
Matthias


------ Originalnachricht ------
Von "JOSEPH SHIPMAN" <joeshipman at aol.com>
An "Foundations of Mathematics" <fom at cs.nyu.edu>
Datum 05.02.2023 16:32:39
Betreff Actual and Potential infinite

>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