Finitism / potential infinity requires the paraconsistent logic NAFL

[Stephen G. Simpson]

Radhakrishnan Srinivasan said:

Simpson [2] argues that potential infinity is justifiable in certain formal
systems that are conservative extensions of the FOL theory PRA.
[...] The FOL theories [...] PRA, PA and stronger theories, do not even
capture the notion of “finite”, in the sense that they admit nonstandard
models.
[...] 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 ).

However, PRA is not an FOL theory (FOL = first-order logic).  PRA is
Skolem's theory of Primitive Recursive Arithmetic, which has number
variables but no quantifiers.   (Skolem's paper is in German, but the
Wikipedia article "Primitive Recursive Arithmetic" gives a useful
summary.)  So PRA contains no quantification over actual infinities, and
anything that can be proved in PRA to exist is finite.  The only infinite
thing about PRA is that the terms, formulas, and proofs form *potential*
infinities.  In my opinion, this is why PRA is widely accepted as a
formalization of Hilbert's idea of pure finitism.

More generally, the various FOL theories that I mentioned (WKL_0, WKL_0 +
RT(2,2), etc.), although they contain actual infinities and even
quantification over actual infinities, are finitistically reducible.
Namely, they are Pi^0_2-conservative over PRA in the following strong
sense: from any proof in these theories of a first-order sentence of the
form "forall x exists y R(x,y)" where R is primitive recursive, one can
primitive recursively extract a primitive recursive function f and a proof
within PRA of R(x,f(x)).

So Radhakrishnan's argument about nonstandard integers seems doubtful, at
least in the case of PRA.  Could Radhakrishnan please explain?

A possible point of confusion: In my book "Subsystems of Second Order
Arithmetic," I discussed PRA as if it were an FOL theory.  My excuse
for this is that I wanted to give a brief discussion of PRA which
would be compatible with the rest of the book, which involves a lot of
actual infinities.

Stephen G. Simpson
Research Professor

Department of Mathematics
1326 Stevenson Center
Vanderbilt University
Nashville, TN 37240, USA

office: 804 Central Library
Vanderbilt University

web: www.personal.psu.edu/t20
email: sgslogic at gmail.com
telephone: 814-404-6176
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