## Number Theory and Infinity Without Mathematics

### Authors

Uri Nodelman and Edward N. Zalta
### Reference

*Journal of Philosophical Logic*, published online 08 August 2024,
doi.org:10.1007/s10992-024-09762-7

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[Authors' preprint available online in PDF]

### Abstract

We address the following questions in this paper: (1) Which set or
number existence axioms are needed to prove the theorems of
‘ordinary’ mathematics? (2) How should Frege’s
theory of numbers be adapted so that it works in a modal setting, so
that the fact that equivalence classes of equinumerous properties vary
from world to world won’t give rise to different numbers at
different worlds? (3) Can one reconstruct Frege’s theory of
numbers in a non-modal setting without mathematical primitives such as
“the number of *F*s” (#*F*) or mathematical
axioms such as Hume’s Principle? Our answer to question (1) is
‘None’. Our answer to question (2) begins by defining
‘*x* numbers *G*’ as: *x* encodes all
and only the properties *F* such that *being-actually-F*
is equinumerous to *G* with respect to discernible objects. We
answer (3) by showing that the mere existence of discernible objects
allows one to reconstruct Frege's derivation of the Dedekind-Peano
axioms in a non-modal setting.