Repository | Journal | Volume | Article

237529

Consistent fragments of grundgesetze and the existence of non-logical objects

Kai Wehmeier

pp. 309-328

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 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.

Publication details

Published in:

(1999) Synthese 121 (3).

Pages: 309-328

DOI: 10.1023/A:1005203526185

Full citation:

Wehmeier Kai (1999) „Consistent fragments of grundgesetze and the existence of non-logical objects“. Synthese 121 (3), 309–328.