, Volume 121, Issue 3, pp 309–328

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


  • Kai F. Wehmeier
    • Philosophical InstituteRijksuniversiteit Leiden

DOI: 10.1023/A:1005203526185

Cite this article as:
Wehmeier, K.F. Synthese (1999) 121: 309. doi:10.1023/A:1005203526185


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 Δ11-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.

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© Kluwer Academic Publishers 1999