[FOM] Disproving Godel's explanation of incompleteness
Jeffrey Ketland
jeffrey.ketland at ed.ac.uk
Wed Oct 19 20:34:29 EDT 2005
Panu,
Roger Bishop Jones wrote:
> Godel believed that:
> (A). The truth predicate of a language cannot be defined in that
> language.
> (B). That (A) explains the incompleteness of arithmetic.
Panu Raatikainen wrote:
> And where exactly Goedel says this ?
See Solomon's Feferman's article "Kurt Goedel: Conviction and Caution"
(1984, reprinted in Feferman 1998, _In the Light of Logic_, pp. 150-164).
Feferman reports the following letter written by Goedel to Arthur W. Burks,
who was editing J. von Neumann's _Theory of Self-Reproducing Automata_
(1966),
::: I think the theorem of mine that von Neumann refers to is not on
::: the existence of undecidable propositions or that on the lengths of
::: proofs but rather the fact that a complete epistemological description
::: of a language A cannot be given in the same language A, because
::: the concept of truth of sentences in A cannot be defined in A. It
::: is this theorem which is the true reason for the existence of
::: undecidable propositions in the formal systems containing arithmetic.
::: I did not, however, formulate it explicitly in my paper of 1931 but
::: only in my Princeton lectures of 1934. The same theorem was
::: proved by Tarski in his paper on the concept of truth published in
::: 1933 in Act. Soc. Sci. Lit. Vars., translated on pp. 152-278 of
::: _Logic, Semantics and Metamathematics_.
See Feferman's article, p. 158.
The von Neumann reference is:
von Neumann, John. 1966: _Theory of Self-Reproducing Automata_. A.W. Burks
(ed.). Urbana: University of Illinois Press.
Best regards --- Jeff
~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
Department of Philosophy
University of Edinburgh
George Square
Edinburgh EH8 9JX
United Kingdom
jeffrey.ketland at ed.ac.uk
~~~~~~~~~~~~~~~~~~~~
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