# Incompleteness in a general setting (vol 13, pg 21, 2007)

Bulletin of Symbolic Logic 14 (1):21 - 30 (2008)
Abstract
Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without getting mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel’s second incompleteness theorem for it. A key role in Löb’s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere’s [7] category-theoretic account of incompleteness phenomena (see also [10]). Incompleteness theorems have also been subjected to intensive investigation within the framework of modal logic (see, e.g.[4], [5]). In this formulation the modal operator takes up the role previously played by the provability predicate, and the derivability conditions on the latter are translated into algebraic conditions (the so-called GL, i.e., Gödel–Löb, conditions) on the former. My purpose here is to present a framework for incompleteness phenomena, fully compatible with intuitionistic or constructive principles, in which the idea of a coding system is retained, only in a 2 simple, but very general form, a form wholly free of syntactical notions. As codes we shall take the elements of an arbitrary given nonempty set, possibly, but not necessarily, the set of natural numbers..
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/4145600
Options
 Save to my reading list Follow the author(s) My bibliography Export citation Find it on Scholar Edit this record Mark as duplicate Revision history Request removal from index

 PhilPapers Archive Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 17,859 External links Setup an account with your affiliations in order to access resources via your University's proxy server Configure custom proxy (use this if your affiliation does not provide a proxy) Through your library Sign in / register and configure your affiliation(s) to use this tool.Configure custom resolver
References found in this work BETA

No references found.

Citations of this work BETA

No citations found.

Similar books and articles

2009-01-28

65 ( #56,011 of 1,781,363 )