Finitism / potential infinity requires the paraconsistent logic NAFL

[Matthias]

A short response to the mail of R. Srinivasan: In the paper you refer 
to, you write comprehensibly

<--
We start with an argument for why the finitist cannot accept the 
existence of nonstandard models of arithmetic, and consequently, must 
reject classical model theory and the existence of infinite sets.
-->

In your mail you write

<--
The FOL theories that these authors use, namely, PRA, PA and stronger 
theories, do not even capture the notion of "finite", in the sense that 
they admit nonstandard models. But the notions of potential infinity 
that these authors embrace ought not to accept nonstandard natural 
numbers as finite.
-->

Indeed, the model theory that I developed is not the classical Tarskian 
model theory. My models are not based on actual infinite career sets, 
but on potential infinite sets (direct systems in case of FOL). So there 
is no completed set N of all natural numbers as a model of PA. There 
exist "nonstandard" numbers, but these are nothing else than 
(indefinitely large) finite numbers. In more detail:

In your paper, you infer from
   (*) (exists m. m = c) and c > n, where n = 0,1,...
that nonstandard models exist. But from a potentialists perspective, the 
formulas in (*) exists only up to some greatest n (not all infinitely 
many 0,1,...), which is a finite number that is not fixed and can be 
arbitrarily large. The set of formulas in (*), roughly {c>0, c>1, ...}, 
is thus not (actual) infinitely large. From that perspective, the 
nonstandard numbers are indefinitely large finite numbers --- for the 
idea of an indefinitely large number see [1], based on [2]. Being 
indefinitely large is a context dependent (hence a relative) notion of 
infinity. When you say 'The finitist recognizes c as an absolute (or 
"completed") infinity', then this is because you treat the set {c>0, 
c>1, ...} as an absolute infinite set. This is not what I do, so your 
criticism does not apply to my approach.

Kind regards,
Matthias

[1] Shaughan Lavine, Understanding the infinite, Harvard University 
Press, 2009. In particular the chapter "The Finite Mathematics of 
Indefinitely Large Size".
[2] Jan Mycielski, Locally Finite Theories, Journal of Symbolic Logic, 
1986.


------ Originalnachricht ------
Von "Radhakrishnan Srinivasan" <rk_srinivasan at yahoo.com>
An fom at cs.nyu.edu
Datum 05.03.2023 15:01:46
Betreff Finitism / potential infinity requires the paraconsistent logic 
NAFL

>In recent papers, [1, 2, 3] the authors have claimed that there exist notions of potential infinity that are justifiable in classical first-order logic (FOL). Linnebo and Shapiro [1] conclude that the Aristotelian notion of potential infinity, which requires modal arithmetic, is correctly captured in the FOL theory PA. Simpson [2] argues that potential infinity is justifiable in certain formal systems that are conservative extensions of the FOL theory PRA. These formal systems are powerful in the sense that they account for much of mathematics, including classical analysis, algebra and geometry. Eberl [3] has developed a model theory that claims to justify potential infinity in FOL theories stronger than PA, which formalize much of mathematics.
>
>In my opinion, all of the above claims concerning potential infinity are flawed, for the following reason. The FOL theories that these authors use, namely, PRA, PA and stronger theories, do not even capture the notion of “finite”, in the sense that they admit nonstandard models. But the notions of potential infinity that these authors embrace ought not to accept nonstandard natural numbers as finite. The potentialist / finitist ought to reject the existence of the nonstandard natural number c defined by the infinitely many axioms c > n (n = 0, 1, ...), because the definition of c requires the existence of an actual (or completed) infinity of standard natural numbers. In other words, for the potentialist / finitist, “finite” ought to mean "standard finite" and hence PA and PRA are inconsistent theories.
>
>I claim that a consistent formulation of finitism is possible in my proposed logic NAFL, which is explained in the following preprint titled "Logical foundations of physics. Resolution of classical and quantum paradoxes in the finitistic paraconsistent logic NAFL":
>
>http://philsci-archive.pitt.edu/21802/
>
>This paper is long (60 pages) but is fairly elementary. In Sec. 1, I explain why the finitist must reject the existence of nonstandard natural numbers, which result from infinitary classical semantics. Sec. 2 defines NAFL semantics and the important metatheorems that follow, including the non-existence of nonstandard models and infinite sets. These metatheorems capture the essence of the NAFL philosophy of finitism. Secs. 5.7 and 5.8 deal with the classical contradictions that result from Zeno's dichotomy paradox (recently discussed in FOM) and their resolution in the NAFL version of real analysis. I also explain why the NAFL truth definition provides a logically consistent explanation for several of the paradoxical phenomena in quantum mechanics.
>
>Comments / suggestions are welcome.
>
>Regards,
>R. Srinivasan
>Independent researcher (retired)
>
>[1] O. Linnebo and S. Shapiro, Actual and potential infinity, Nous 53, No. 1 (2019) 160–191.
>
>[2] S. G. Simpson, Foundations of Mathematics: an Optimistic Message. In: R. Kahle, M. Rathjen (eds) The Legacy of Kurt Schutte (Springer, Cham. (2020) https://doi.org/10.1007/978-3-030-49424-7_20 ).
>
>[3] M. Eberl, A model theory for the potential infinite, Rep. Math. Log. 57 (2022) 3–30.