FOM: Re: Goedel: truth and misinterpretations

[Jeffrey Ketland]

Here's a summary:

Ketland:
> 5. Does mathematical truth mean "having a mathematical proof"? (Kanovei)
> Obviously NOT. The truth set for (|N, 0, s, +, x) is not axiomatizable.
This
> was discovered by Kurt Goedel in 1930. N.B. The definition (v) of truth
(for
> sentences of the lang of set theory) above makes no mention of *proof*.

Kanovei:
>This is probably the most splendid misinterpretation of any
>mathematical result, that persists 70 years in discussions
>between clones of philosophers (and mathematicians, and, shamefully,
>logicians, too).
>Goedel proved (in a very weak fragment of ZFC, essentially, in PA)
>that for any theory T of certain kind, if Cons T then there is an
>arithmetical statement A such that neither Prov_T A nor Prov_T (not-A),
>where Cons and Prov express the consistency and provability (in T).
>Then Ketland notes: gee, either A or not-A is true, hence, there is
>something true but not provable.

A splendid misinterpretation!

Start again. Using standard methods, we define (in ZFC or Z_2 say) an
adequate truth predicate for arithmetic. OK? (Forget a truth predicate for
lang of set theory - sure, we need a theory of classes like MK or a
primitive satisfaction predicate). The following sentences are then
*theorems* of ZFC and of Z_2 (second-order arithmetic) as well:

(1) For any sound recursively axiomatized formal system F, there are true
sentences of arithmetic not provable in F.

(2) The set of arithmetic truths is not r.e.

By *Kanovei's criteria* (provability in ZFC), (1) and (2) must be
mathematical
facts. But these mathematical facts contradict his *philosophical* position!
(Since the deductive closure of a formula system is r.e., i.e., Sigma_1).

On my own realist view, they are facts too: unless something extremely weird
happens to the evolution of human ideas, they will be as widely accepted by
the mathematical community in a billion years, as they are today. Just like
the irrationality of the square root of 2. Mathematical truth is NOT the
same as provability in a formal system.

I would argue that these mathematical discoveries by Goedel and Tarski point
strongly in the following philosophical direction. Statements can be true
without an "observer" or a proof. If it weren't for some very spooky facts
about the measurement problem in quantum theory, I would feel secure in
saying that reality doesn't pop into existence when you "observe" it or
prove that something is the case.

Question for Kanovei (or anyone else): do you think that the standard
definition of the truth set for (|N, 0, s, +, x) doesn't make sense?

(Because this set CANNOT be the deductive closure of ANY (consistent) formal
system).

Furthermore, I am not unique in my misinterpretations: E.g.,

------------------------
W.V. Quine 1978 (obituary of Goedel) wrote:

Elementary number theory is the modest part of mathematics that is concerned
with the addition and multiplication of whole numbers. Whatever sound and
usable rules of proof one may devise, some truths of elementary number
theory will remain unprovable; this is the gist of Gödel’s theorem. 

We used to think that mathematical truth consisted in provability. Now we
see that this view is untenable for mathematics as a whole, and even for
mathematics in any considerable part; for elementary number theory is indeed
a modest part, and it already exceeds any acceptable proof procedure.
(Quine 1978 in Theories and Things 1981, p. 144).

------------------------
Also, Boolos and Jeffrey 1989 (textbook "Computability and Logic") write:

And perhaps the most significant consequence of Theorem 6 [Gödel’s First
Incompleteness Theorem] is what it says about the notions of truth (in the
standard interpretation of the language of arithmetic) and theoremhood, or
provability (in any particular formal theory): that they are in no sense the
same.
(Boolos & Jeffrey 1989, p. 180).

---------------------

So, at least I'm in good company! ; )

Regards - Jeff

~~~~~~~~~~~ Jeffrey Ketland ~~~~~~~~~
Dept of Philosophy, University of Nottingham
Nottingham NG7 2RD United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~