Finitism / potential infinity requires the paraconsistent logic NAFL

[Radhakrishnan Srinivasan]

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.