Godel's Theorem and truth
David Libert
all started when someone phrased Godel's first incompleteness theorem as
producing G(T), the Godel sentence for T, as true but unprovable, and then
asked what true means in this context.
Godel already anticipated this sort of problem, and worked around it in
the original 1931 paper.
I will quote from _Kurt Godel Collected Works Volume 1 1929-1936_
edited by Solomon Feferman et al.
Stephen C. Kleene in that volume has an introductory note for Godel's
paper and several related papers by Godel from this time. Writing about
Godel's main 1931 paper Kleene writes:
" In Section 1 of 1931, Godel undertakes to "sketch the main of the
proof [of his 'first incompleteness theorem'], of course without any
claim to complete precision." This corresponds to the foregoing
preliminary outline; but he actually sketches the construction of the
formula A (the "S" and "A" here are "PM" and "[R(q);q]" in his Section
1).
In the foregoing sketch, the interpretation of the formal system as
formalizing a system of notions and propositions when we talked of
constructing a formula A or [R(q);q] which should express its own
unprovability and assumed that only true formulas are provable in S or
PM. Yes, one wants this to be so, to the extent that we can be
satisfied that the formulas have clear meanings. But the informal
concept of truth was not commonly accepted as a definite mathematical
notion, especially for systems like PM and Zermel-Fraenkel set theory.
In Hilbert's metamathematics one is supposed to deal solely with the
forms of formulas, using only finitary reasoning.
So, near the end of Section 1, Godel says, "The purpose of carrying
out the above proof with full precision in what follows is, among other
things, to replace [the assumption that every provable formula is true
in the interpretation considered] by a purely formal [or
metamatematical] and much weaker one. "
So in the 1931 paper _On formally undecidable propositions in Principia
mathematica and other related systems I_ Godel has a Section 1, of 3.5
pages in my edition. This section gives an outline of the coming proof,
and is phrased in terms of truth, it being assumed that the underlying
system only proves true statements, in other words a sort of consistency
assumption. But this does not claim to be a complete a proof, rather an
idea for a proof.
The proper proof of the first incompleteness theorem is given in Sections
2 and 3 which follow. As Godel notes at the end of Section 1, the Section
1 discussion was only a leadin to the real proof of these later sections.
Sections 2 and 3 together span 40 pages in my edition. In these sections
no notion of truth is assumed or used. And the theorem proved is not of
the form from the thread as I mentioned above. Instead it is all phrased
in terms of omega-consistency, as quoted above a weaker metamathematical
notion than truth.
I recall having heard a comment, but I have not been able to track it
down. Namely Godel at this time considered that truth was not such a
questionable notion, but given the time he felt his proof would be more
accpeted if it did not depend on such notions. So he wrote the 1931 paper
in a very formal style, working with low level notions, to make it more
readily accepted.
Along these lines, I was told in conversation once by reputable source
that with the Liar, Richard and Berry paradoxes known from the early 1900's
or earlier, in the 1930's there was a general distrust of truth as an
abstract notion among philosophers. Tarski intended to counter this, with
his work on truth. This work was in two parts. Tarski showed that in
certain cases truth is undefinable. But on the flip side he gave a
definition of truth in other cases. The part showed that used with proper
restraint it was a proper notion.
I think this Tarski work was published around 1936. So Godel was working
in the context before that.
Popularizers of Godel since often phrase things again in terms of truth,
leaving open the kind of discussions as in the other threads. A better
version is as from the main theorem of Godel.
By the way, Section 1 does not state a definite theorem. It just
indicates itself as giving a general outline of the later sections, and the
actual theorems of Sections 2 and 3 don't involve truth.
Namely this version of the first incompleteness theorem says that for
recursively axiomatized theories T interpreting enough arithmetic (PA in
Godel's original version, Robinson's arithmetic in later improvements),
there is the Godel sentence G(T), s.t. if T is consistent then T does not
prove G(T), and if T is omega-consistent T does not prove ~G(T).
--
David Libert (ah...@freenet.carleton.ca)
1. I used to be conceited but now I am perfect.
2. "So self-quoting doesn't seem so bad." -- David Libert
3. "So don't be a morron." -- Marek Drobnik bd308 rhetorical salvo IRC sig
Torkel Franzen
> I recall having heard a comment, but I have not been able to track it
> down. Namely Godel at this time considered that truth was not such a
> questionable notion, but given the time he felt his proof would be more
> accpeted if it did not depend on such notions. So he wrote the 1931 paper
> in a very formal style, working with low level notions, to make it more
> readily accepted.
On this, see Feferman, S., Kurt Gödel: Conviction and Caution
(1983), in S.G. Shanker (ed.), Gödel's Theorem in Focus, Croom
Helm 1988.
Torkel Franzen
> On this, see Feferman, S., Kurt Gödel: Conviction and Caution
> (1983), in S.G. Shanker (ed.), Gödel's Theorem in Focus, Croom
> Helm 1988.
Also to be found in his collection _In the Light of Logic_.