Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

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

Kai F. Wehmeier
Synthese
Vol. 121, No. 3 (1999), pp. 309-328
Published by: Springer
Stable URL: http://www.jstor.org/stable/20118232
Page Count: 20
  • Download ($43.95)
  • Cite this Item
Consistent Fragments of "Grundgesetze" and the Existence of Non-Logical Objects
Preview not available

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.

Page Thumbnails

  • Thumbnail: Page 
[309]
    [309]
  • Thumbnail: Page 
310
    310
  • Thumbnail: Page 
311
    311
  • Thumbnail: Page 
312
    312
  • Thumbnail: Page 
313
    313
  • Thumbnail: Page 
314
    314
  • Thumbnail: Page 
315
    315
  • Thumbnail: Page 
316
    316
  • Thumbnail: Page 
317
    317
  • Thumbnail: Page 
318
    318
  • Thumbnail: Page 
319
    319
  • Thumbnail: Page 
320
    320
  • Thumbnail: Page 
321
    321
  • Thumbnail: Page 
322
    322
  • Thumbnail: Page 
323
    323
  • Thumbnail: Page 
324
    324
  • Thumbnail: Page 
325
    325
  • Thumbnail: Page 
326
    326
  • Thumbnail: Page 
327
    327
  • Thumbnail: Page 
328
    328