[FOM] Godel Sentence & The Liar

[Arnon Avron]

> 
> 
> Let's say the Godel Sentence is P, and P says "P is not provable".
> Historically, the Godel Sentence is often said to be derived from, or maybe
> just related to,  the Liar.
> 
> While the Liar is usually rendered as "This sentence is false.", I have seen
> a variant something like "This sentence cannot be proven true.", and I can't
> think of a reason why it couldn't also appear as something like "This
> sentence cannot be verified (as true(?))."
> 
> My question is, does the fact that the Godel Sentence is supposed to be true
> but unprovable suggest a possible solution to The Liar?  I have read
> accounts where the Liar is taken to be false, or perhaps without truth
> value, or perhaps both true and false.  I have not come across an account
> where it is interpreted as true but unprovable/unverifiable.  Isn't there an
> analogy to be exploited here, or not?  Wouldn't the existence of true but
> unverifiable propositions be an interesting result?


To my opinion, there is no analogy whatever. The sentence 
"This sentence cannot be proven true" immediately leads to a 
paradox if we assume (wrongly, I am convinced) that it expresses
a meaningful proposition. Godel sentences are not paradoxical at all.
The difference is that "This sentence cannot be proven true"
would have been about itself (and only about itself) had it
been meaningful, while a Godel sentence is not about itself.

The whole problem here is based on a misunderstanding. Unfortunately,
this misunderstanding is based not only on popular, non-technical 
presentations of Godel's proof, but also on careless statements
that can be found even in advance textbooks and papers (recently I saw 
such a careless statement in Girard's paper in the BSL!). This is
a good opportunity to warn about this carelessness. So let me repeat:
A Godel sentence does NOT "say about itself that it is unprovable".
In fact it does not say anything at all about itself. A Godel Sentence
is a sentence in the language of elementary arithmetics that expresses
a certain property of addition and multiplication of natural numbers.
If somebody presents to us a Godel sentence without telling us
that it is a Godel sentence, and asks us whether it is true or not,
we most probably would not even guess that it is a Godel sentence-
but it will still be meaningful to us, no more and no less then any
other Pi-0-1 arithmetical sentence. It is only through some particular
method of coding that a Godel sentence can be interpreted as
"talking about itself" (which means: conveying a property of a certain
natural number, that can be made the code of that very sentence
according to some,  very special method of coding). Note however
that every text can be used to convey whatever information we like
by using some method of coding!  Thus in Israel it became
popular in recent years to see the old testimony as a huge message
by god, written in a code (called "the biblical code") in which one
can find information about everything in the past, present, and future
(This code has proved to have  practical applications: some people 
have indeed made a lot of money by writing books about it!). 
Now the fact that a certain text has been  given a certain interpretation
through coding does not change the meaning of that text according to the
language in which it is written - and it is irrelevant here whether 
that text was apriorily written with a particular code in mind,
and in order to be used for delivering something different than its
actual content, or whether such a code was found later, independently
of the original intention of the person who had produced that text
(this is the case at least for declarative texts which are not dependent on
time or context, like mathematical texts). So again: a Godel sentence
for itself is an arithmetical sentence. No more. Using different 
methods of coding it may (like any other sentence) be used to indirectly
express all types of facts, including many which have nothing to do
with the natural numbers. What is special of a Godel sentence is that
it has one special interpretation the subject of which is the sentence
itself, and this interpretation (here come the ingenuity of Godel!)
can be used to show that this sentence is true (and not provable
in a certain theory).

This brings me to another common point of carelessness. I have talked
all the time about A Godel sentence, not about THE Godel sentence. The
reason is simple: there is no such thing as "THE Godel sentence".
What is used as a Godel sentence in Kleene's book (say) is completely
different from what is used as a Godel sentence in Smullyan's book.
The identity of "The Godel sentence" depends on the numbers which
the primitive symbols are assigned (and on the choice of the
primitive symbols), on the way sequences (or maybe trees?) are codified,
on the proof system which is used (Hilbert? Gentzen? ND?) and dozens
of other factors. Make any change and "the Godel sentence" stopped
to be "the Godel sentence" (this would not change of course its
content or truth-value!). Of course, one may claim that all 
Godel sentences are equivalent. This claim is trivially true 
(and insignificant) if by the equivalence of A and B one means
that A<->B is a true sentence. So a more sophisticated notion
of equivalence is needed here. I have indeed no doubt that under 
certain conditions concerning the codification methods, two 
Godel sentences for the same theory are provably equivalent
at least in that theory (and maybe a weaker one). But I doubt that
one may find conditions which are general enough to cover any
sentence that one might call in certain circumstances
"a Godel sentence for the theory T". Can anyone give me references
to works which have been done on this issue? (note that the equivalence
in T of a Godel sentence for T to Consis_T is not helpful here: Consis_T
is no more uniquely defined than "The Godel sentence of T").

 This brings me to another question. How exactly should the notion
of a Godel sentence for a theory T be defined, and is there an
effective (or at least independent of the way it has been constructed)
criterion for identifying Godel sentences? 
As far as I know (and PLEASE correct me if I am wrong) it is quite
possible that there exists a Godel  sentence for PA which we can 
easily see to  be true iff Goldbach conjecture is true - but it is not 
so easy to see (if at all) that it is in fact a Godel  sentence for PA.
Again: does anybody know about works which were done in this direction?

Arnon Avron