Zoom Talk: Ed Zalta (Senior Research Scholar, Stanford)
Co-author: Uri Nodelman (Senior Research Engineer, Stanford)
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Number Theory Without Mathematics
No specifically mathematical primitives or axioms are required to derive second-order Peano Arithmetic (PA2) or to prove the existence of an infinite cardinal. We establish this by improving and extending 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 improve the results by developing a Fregean approach to numbers that accomodates a modal setting, yielding numbers that are stable across possible worlds, even though the equivalence classes of equinumerous properties vary. To extend the results, we (a) prove a Recursion theorem (which shows that recursive functions are relations grounded in second-order comprehension), (b) derive PA2, and (c) re-derive the existence of an infinite cardinal and (d) derive the existence of an infinite set (where sets are defined non-mathematically as extensions of properties). Since the background framework of object theory has no mathematical primitives and no mathematical axioms, we have a mathematics-free foundation for number theory.