, Volume 121, Issue 3, pp 309-328

Consistent Fragments of Grundgesetze and the Existence of Non-Logical Objects

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Abstract

In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory TΔ in monadic second-order logic, consisting of axiom V and Δ1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck and TΔ prove the existence of infinitely many non-logical objects (TΔ deriving,moreover, the nonexistence of the value-range concept). Some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed. Finally, I explore the relation of TΔ toCantor's theorem which is somewhat surprising.