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TWO-SORTED FREGE ARITHMETIC IS NOT CONSERVATIVE

2022; Cambridge University Press; Volume: 16; Issue: 4 Linguagem: Inglês

10.1017/s1755020322000156

1755-0203

Stephen Mackereth, Jeremy Avigad,

Logic, Reasoning, and Knowledge

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.