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

Author: Wehmeier K.F.

Source: Synthese, Volume 121, Number 3, December 1999 , pp. 309-328(20)

Publisher: Springer

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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 T_Δ in monadic second-order logic, consisting of axiom V and Delta ^1_1-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 Heck and T_Δ prove the existence of infinitely many non-logical objects (T_Δ 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 T_Δ to Cantor's theorem which is somewhat surprising.

Language: English

Document Type: Regular paper

Affiliations: 1: Philosophical Institute Rijksuniversiteit Leiden Matthias de Vrieshof 4 Postbus 9515 2300 RA Leiden The Netherlands E-mail: wehmeier@rullet.leidenuniv.nl

Publication date: 1999-12-01