, Volume 121, Issue 3, pp 309–328 | Cite as

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

  • Kai F. Wehmeier


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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Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Kai F. Wehmeier
    • 1
  1. 1.Philosophical InstituteRijksuniversiteit LeidenLeidenThe Netherlands

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