Re: Godel's comments about the "true reason" for incompleteness

On Mar 22, 7:40 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
On 2008-03-22, in sci.logic, R. Srinivasan wrote:

I do not agree that this is a straghtforard process.

You're of course free to disagree with anything you like, but unless
you provide some specific reason for your disagreement it remains
entirely arbitrary and uninteresting. Do you, in fact, envisage some
specific difficulty in formalising talk about formal sentences?


Since you are the Godel expert, I will take this opporunity to sound
you out on the discussion I have been having with MoeBlee and a
possible paradox, of which I will give a high-level description here.

In your previous post, you said:

\begin{quote}
We just need to choose a language in which statements
about sentences are directly expressible. There's nothing problematic
in this, and the most obvious choices for a theory expressing natural
principles regarding formal sentences result in theories synonymous,
in the technical sense, with (subtheories of) Peano arithmetic.
\end{quote}

Taking this as the cue, I want to set up two theories which, in
prinicple ought to be entirely equivalent to each other, but which
aren't -- hence the paradox. Consider the theory PA and, by what you
have said above, an entirely equivalent theory T in which we are able
to formulate statements about sentences of PA. Both T and PA are first
order theories. We now perform a mutual translation as follows. Using
Godel's procedure, let us translate all notions about sentences of PA
expressible in the language of T into the language of PA. Secondly, we
also translate all arithmetical notions expressible in PA into the
language of T.

Now consider the following sentence of T:

(Fot all P) (For all Q) (P&~P)->Q (*)

where P and Q range over sentences in the language of PA.

(*) is a sentence expressible in the language of T. What (*)
expresses is that "from any contradiction in the language of PA, an
arbitrary sentence of PA will follow". Further, let (*) translate to
the arithmetical proposition S in the language of PA. Now my
understanding is that there is such a sentence S and that

PA |- S (**)

My dispute with MoeBlee was that S *ought* to be undecidable in PA.
Here is why I think so. The sentence (*) (which can be thought of as
the inverse translation of S into T) has to be undecidable in T by my
reckoning, since the arithmetical ;principles of PA translated into T
will not be able to either prove or refute (*). Why is this? Because
we know that PA does not even recognize (*) as a legitimate sentence;
PA can only recognize and prove the infinitely many instances of (*)
(corresponding to specific P and Q), but not (*) itself. We have only
translated the principles of PA into T, so my argument is that (*) is
undecidable in T. However, we find by (**) that S is provable in PA.
In fact there is no way tor S to be undecidable in PA, for that would
lead to a contradiction. Similarly the undecidability of (*) in T
would also lead to contradiction if we insist, by the completeness
theorem, that there must exist a model of T in which (*) is false.

Thus we have the strange situation in which two supposedly equivalent
theories PA and T are not equivalent after all. What do you think
about this? Does this argument establish that S ought to be
undecidable in PA as I have been arguing with MoeBlee? In which case
we have to conclude that the translation procedure is illegitimate.
Alternatively, I think that this argument shows why we are *compelled*
to allow (*) as a legitimate propostion of PA, assuming we accept
Godel's translation procedure.


Obviously you are now claiming that a single sentence in the
language of theory can say something about all sentences in the
language.

That is, in fact, just what I am claiming.


Please see my argument above for why I think this situation leads to
paradoxes.


In first order logic, sentential variables are not part of sentences
and there is a valid reason for that as far as I am concerned.

In the straightforward formalisation of talk about formal sentences I
alluded to there is no need to involve sentential variables.

OK, but you do need to quantify over the formal sentences.

Regards, RS
.

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