Mathematics with the potential infinite

[Matthias]

Dear Tim,

my approach is different in the sense that I want to understand (almost) 
all of mathematics with the notion of a potential infinite. So I adopt 
model theory and let the calculus (e.g. classical logic) basically as it 
is --- in other words, there should be no statements that become 
unprovable using the potential infinite. I say "basically", because 
there is a restriction in what can be *formulated* in a way that makes 
sense from the point of view of a potential infinite. Problematic is in 
my opinion the totality of higher-order application, which I already 
mentioned (this is also related to the discussion in some of Weyl's 
writings).

Different to completed infinite sets in set theory, I use systems, since 
they are "dynamic" (roghly, direct system for finite objects, inverse 
systems for infinite objects). One might say that within all of these 
systems (used in an investigation), you never do the final limit step, 
you always stay within the system. So you keep track of the stages of 
each of these systems, in particular keep track of how far you have to 
go to be sound. It might be not decidable which stages are required in 
some cases, though, but they exist from a finitistic perspective. In 
other words, I start with a basic finitistic principle, casually 
formulated

(FP0) Every object lives within a finite world.

For instance, each natural number n occurs in some finite set N_i = {0, 
..., i-1}, clearly, take any i>n. The same holds for expressions: each 
formula is within a finite set of expressions that I use for my 
investigation. Now extend this principle to other mathematical objects. 
E.g. each property/set on natural numbers exists within some finite set 
P(N_i) --- here one needs inverse systems instead of direct systems, 
otherwise one considers only fixed finite sets, no potential infinite 
sets. And then apply it to functions as well.

Again formulated differently, you use the reflection principle from the 
very beginning. In ZFC set theory it is usually applied at the end. So 
you allow unrestricted quantification over all (indefinitely extensible 
stages of the set theoretic hierarchy (V_\alpha)_{\alpha an ordinal}) 
and at the end say, roughly, for a finite set of formulas, we could have 
used some stage V_\alpha instead of V. Similar for Grothendieck 
universes in models of type theory, where at the end one creates a 
hierarchy of universes up to \omega (or 2*\omega, in Agda, as far as I 
know).

I am convinced that you have to apply a reflection principle in order to 
interpret the universal quantifier in a potential infinite world. A 
naive interpretation boils down to assume a definite extension of the 
concept "natural number", which means an actual infinite set N. The 
question is: What should it mean to see N as potential infinite, and at 
the same time talk about all natural numbers?

Regarding the FOM discussion as of 2015 (I probably joined later). I 
agree with Arnon Avron, when he stated that
<--
I take this opportunity to note about the false claim that quantifying 
over N in general, and PA in particular, are justified only if one 
accepts N as a complete object which is actually infinite.
According to this "logic", the quantification that is done in ZF over 
the universe of sets is justified only if one accepts V as a complete 
object which is absolutely infinite.
-->

and I agree with Marcin Mostowski, saying that
<--
Timothy Y. Chow observed:
I've never seen anyone define two separate axioms and declare one of 
them to be an "axiom of potential infinity" and the other an "axiom of 
actual infinity."

No wonder, potential and actual infinity are [not?] views on 
mathematical truths, but on nature of mathematical reality.
-->

In the same direction, i.e., there is no axiomatic way to express what 
potential infinte means, goes the analysis of Niebergall, mentioned in 
my paper.

Marcin Mostowski's approach however leads to a restriction, which 
functions and relations are allowed within a potential infinite reality. 
His approach is less dynamic than Jan Mycielski's. Mycielski allows that 
within a single formula the range of quantifier may change. This is in 
line with Dummett's notion of "indefinite extensibility" and with 
reflection principles. However, Mycielski uses a translation of formulas 
and then applies the common Tarskian model theory, whereas I do not 
manipulate expressions.

Regards,
Matthias


------ Originalnachricht ------
Von "Timothy Y. Chow" <tchow at math.princeton.edu>
An fom at cs.nyu.edu
Datum 26.01.2023 01:14:59
Betreff Re: Mathematics with the potential infinite

>Matthias wrote:
>
>>for several years I have been engaged in developing mathematics solely with the concept of the potential infinity. Two publications of mine are now available.
>
>This is interesting.  The topic of potential infinity has come up on FOM before a few times, e.g., in March 2015---
>
>https://cs.nyu.edu/pipermail/fom/2015-March/
>
>---and in July 2020, when you made a few comments:
>
>https://cs.nyu.edu/pipermail/fom/2020-July/
>
>One question I have in general for any account of potential infinity is this: Can you give examples of statements that are provable using "actual infinities" but that are not provable using "potential infinity" only? (Here, I exclude statements that don't even make sense unless you assume actual infinities.)  As a specific test case of interest, can you prove "PA is consistent" using only only potential infinity?
>
>Tim