Three theorems of Godel


It might seem that three of Godel’s results - the Completeness and the First and Second Incompleteness Theorems - assume so little that they are reasonably indisputable. A version of the Completeness Theorem, for instance, can be proven in RCA0, which is the weakest system studied extensively in Simpson’s encyclopaedic Subsystems of Second Order Arithmetic. And it often seems that the minimum requirements for a system just to express the Incompleteness Theorems are sufficient to prove them. However, it will be shown that a particular sub-system of Peano Arithmetic is powerful enough to express assertions about syntax, provability, consistency, and models, while being too weak to allow the standard proofs of the theorems to go through. An alternative proof is available for the First Incompleteness Theorem, but is of such a different nature that the import of the theorem changes. And there are no alternative proofs for (certainly) the Completeness and (apparently) the Second Incompleteness Theorems. It is therefore perfectly rational for someone to be skeptical about Godel’s results.



    Upload a copy of this work     Papers currently archived: 89,764

External links

  • This entry has no external links. Add one.
Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

  • Only published works are available at libraries.

Similar books and articles

Recursion theory for metamathematics.Raymond Merrill Smullyan - 1993 - New York: Oxford University Press.
Gödel's incompleteness theorems and computer science.Roman Murawski - 1997 - Foundations of Science 2 (1):123-135.
Gödel's incompleteness theorems.Raymond M. Smullyan - 1992 - New York: Oxford University Press. Edited by Lou Goble.
Incompleteness in a general setting (vol 13, pg 21, 2007).John L. Bell - 2008 - Bulletin of Symbolic Logic 14 (1):21 - 30.
Metamathematics, machines, and Gödel's proof.N. Shankar - 1994 - New York: Cambridge University Press.
On the philosophical relevance of Gödel's incompleteness theorems.Panu Raatikainen - 2005 - Revue Internationale de Philosophie 59 (4):513-534.


Added to PP

584 (#26,371)

6 months
1 (#1,018,209)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references