Two-sorted Frege Arithmetic is not Conservative

Review of Symbolic Logic:1-34 (2022)
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Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding 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.

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Author Profiles

Jeremy Avigad
Carnegie Mellon University
Stephen Mackereth
University of Pittsburgh

Citations of this work

Definitions.Anil Gupta - 2008 - Stanford Encyclopedia of Philosophy.

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