Talk: Ed Zalta (Senior Research Scholar, Stanford University)

Co-Author:
Uri Nodelman (Senior Research Engineer, Stanford University)

Title:

Number Theory Without Mathematics Redux

Abstract:

Last year, in a presentation to MCMP via Zoom, we showed that no mathematical primitives or axioms are required to derive 2nd-order Peano Arithmetic (PA2) or to prove the existence of an infinite cardinal. In that talk, we improved and extended the results of Zalta 1999 ("Natural Numbers and Natural Cardinals as Abstract Objects", J. Philosophical Logic, 28(6): 619-660), in which the Dedekind-Peano axioms for number theory were derived in an extension of object theory. We improved Zalta 1999 by deriving a single group of numbers stable across possible worlds, even though the equivalence classes of equinumerous properties vary across worlds. And we extended Zalta 1999 by proving a Recursion theorem and deriving the other axioms of PA2 as well as an infinite cardinal. But we were still restricted to counting only ordinary objects and, in deriving (a) the axiom that every number has a successor and (b) the claim that there exists an infinite cardinal, we relied upon one of the *modal* axioms of object theory. Moreover, the derived infinite cardinal wasn't provably aleph_0. Over the past year we've discovered that if we redefine and recast our Frege-style number theory so that "discernible" objects can be counted (including some abstract objects along with ordinary objects), the modal axiom isn't needed to to derive (a) and (b), and the infinite cardinal that becomes derivable is in fact aleph_0. So in this talk, we show how the work we presented last year can be preserved and strengthened even if we start with one less axiom, thereby resulting in an improved, mathematics-free foundation for number theory.