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

Kai F. Wehmeier
Synthese
Vol. 121, No. 3 (1999), pp. 309-328
Stable URL: http://www.jstor.org/stable/20118232
Page Count: 20
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Abstract

In this paper, I consider two curious subsystems of Frege'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 $\text{T}_{\Delta}$ in monadic second-order logic, consisting of axiom V and Δ₁ⁱ-comprehension (in a language containing an abstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both H and $\text{T}_{\Delta}$ prove the existence of infinitely many non-logical objects ($\text{T}_{\Delta}$ 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 $\text{T}_{\Delta}$ to Cantor's theorem which is somewhat surprising.

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