Two-Sorted Frege Arithmetic is Not Conservative

Review of Symbolic Logic 16 (4):1199-1232 (2022)
  Copy   BIBTEX

Abstract

Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. In fact, 2FA is not conservative over n-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic.

Author Profiles

Stephen Mackereth
University of Pittsburgh
Jeremy Avigad
Carnegie Mellon University

Analytics

Added to PP
2022-04-11

Downloads
215 (#83,307)

6 months
160 (#22,226)

Historical graph of downloads since first upload
This graph includes both downloads from PhilArchive and clicks on external links on PhilPapers.
How can I increase my downloads?