Skip to main content
    • Aa
    • Aa

Calculating self-referential statements: Guaspari sentences of the first kind

  • C. Smoryński (a1)

Beginning in 1960, and continuing for about a decade and a half, Shepherdson's self-referential formulae dominated the applications of diagonalization in metamathematics. In 1976, it was doubly toppled from its position of supremacy by two demonstrably more powerful such sentences introduced by D. Guaspari. The goal of the present note is to understand and explain the first (and, for the time being, more important) of Guaspari's self-referential sentences, which we dub “Guaspari sentences of the first kind”, or, less poetically, “Guaspari fixed points”.

Both Shepherdson's and Guaspari's fixed points are generalizations of the Rosser sentence. But, where Shepherdson merely tacks on side-formulae, Guaspari takes a more revolutionary step: He views the basic components, PrT(⌈¬φ⌉) and PrT(⌈φ⌉), of the Rosser sentence as attempts to refute something and replaces them by refutations of something else. Ignoring their Shepherdsonesque side-formulae (tacked on for the sake of more esoteric applications), our analysis of Guaspari fixed points can be viewed as merely the isolation of those properties whose refutations can be used in this context.

In §1 we offer the outcome of this analysis—i.e. a delineation of properties whose refutations can so be used. Some simple examples are cited and the Guaspari fixed points are defined. The main theorem—a master Fixed Point Calculation—is proven in §2.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] D. Guaspari , Partially conservative extensions of arithmetic, Transactions of the American Mathematical Society, vol. 254 (1979), pp. 4768.

[5] J. Shepherdson , Representability of recursively enumerable sets in formal theories, Archiv für Mathematische Logik und Grundlagenforschung, vol. 5 (1960), pp. 119127.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 1 *
Loading metrics...

Abstract views

Total abstract views: 41 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd May 2017. This data will be updated every 24 hours.