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

On Mar 22, 8:56 pm, george <gree...@xxxxxxxxxx> wrote:
On Mar 21, 9:18 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

The main point to which we both agree on is that a first-order
language cannot express sentences like "All sentences follow from a
contradiction".

This is simply false.

The infinitely many sentences in this schema may all
be tautologies proven by the object theory, but the object theory
cannot formalize the notion of "all sentences", at least not without
the coding employed by Godel.

This reflects a complete MISunderstanding of what
ANY sentence in a classical formal first-order language
IS EVER "about".

The fundamental simple answer is that no first-order
theory IS EVER about ANYthing.

The way that the canonical paradigm handles SEMANTICS is
that you get to interpret your first-order language *IN ANY* way
you like. *NO* particular interpretation is privileged.
EVERY interpretation is (at step 0) psychologically (and logically)
AVAILABLE. Going from step 0 to step 1, the first step, requires
excluding all interpretations THAT FALSIFY some sentence of
the theory. At step 1, ALL interpretations, about WHATEVER
realms of discourse you can imagine, REMAIN available.
Unless the theory is categorical, some of these interpretations
are going to assign DIFFERENT meanings to some functors
and predicates! Yet the theory, viewed purely as a collection
of sentences, IS SIMULTANEOUSLY "about" ALL (and therefore
none) of them!

The point is, since the elements of your domain
CAN BE ANYthing, since your first-order quantifiers
can quantify OVER *ANY* class whatSOEVER, they may,
in particular, quantify over the class of sentences in your
language. I think your next question is going to be about
how, if you're not careful, this doesn't automatically promote
you to 2nd-order. Obviously, there is a sense in which IT DOES,
but equally obviously, as long as we have set theory, we can just
ignore that. Encoding will be relevant but it will not be complicated
and it will not prevent the theory from remaining *formally* first-
order. For the theory to be FORMALLY 2nd-order, there have
to be SYMBOLS IN THE LANGUAGE (as OPPOSED to elements
in the interpretation) that are SYNTACTICALLY allowed to occur
as *both* first-order predicates AND arguments to 2nd-order predicates
and functions. Under the classical semantics for 2nd-order, it is
also
necessary for ALL THEORETICALLY POSSIBLE subcollections of the
first-order domain to be available as elements-quantified-over by the
2nd-order quantifiers.

Since 1st-order treatments that "quantify over all sentences" are NOT
going to meet EITHER of these constraints, they are NOT (formally)
2nd-order.


Suppose, as you say, it is indeed legitimate to interpret the
quantifiers in a ifrst-order language, say that of PA, as quantifying
over the sentences in the language.. Obviously you need some kind of
coding to do this, and let us also accept Godel's translation
procedure as legitimate.Here is a step-by-step outline of the
difficulties that would follow:

1. Godel first costructed a metatheory T of PA in whose language we
may construct and prove propositions about sentences of PA.

2. The theory T is, in substance, precisely as strong as PA. I.e., we
may encode PA into the theory T, and nothing stronger than PA.

3. Godel used his translation procedure to translate the theorems of T
into the language of PA.

4. In particular, consider the following sentence in the language of
T:

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

5. Let us say that (*) gets translated to the sentence S in the
language of PA.

6. It turns out that S is provable in PA, as per my understanding.

7. However, the theory T is precisely as strong as PA, which does not
prove the sentence (*).(PA only proves all instances of (*), but not
(*) iitself). Therefore T shoudl not prove (*). T cannot refute (*)
either, without proving some contradiction. The undecidability of (*)
is T is also problematic, since there cannot exist a model of T in
which (*) is false. So apparently we have a problem here, since T is a
first-order theory for which the completeness theorem must hold. If T
were a second order theory, this difficulty would presumably not have
arisen and we could accept undecidability of (*) in T.

8. My argument is that S should be similarly undecidable in PA, since
it is a direct translation of the sentence (*) and T does not either
prove or refute (*). But such undecidability would also be
problematic.

9. Note that one could possibly impose a requirement like "If PA
proves all instances of a schema, then PA also proves the schema
itself (with sentential variables in it). Such a requirement would
justify the provability of S in PA and of (*) in T. But such a
requirement would also be inconsistent with the results of Godel's
incompleteness theorems.

10. Looks like the only other way out is to reject the translation
procedure of Godel as illegitimate.


The
claim that such a meta-sentence can be translated into a sentence in
the object language is what iI am contesting. Such a translation
cannot be via purely finitary first-order reasoning.

NOTHING IS EVER *purely* finitary in first-order logic.
At a bare minimum, the language itself is denumerably
infinite. More substantively, the overall domain of discourse
has to be infinite -- OTHERWISE YOU MIGHT AS WELL
BE USING *propositional/0th-order* logic IN THE FIRST PLACE.



Now, perhaps you can move on also to disabuse yourself of the notion
of adding just a single sentence to a language.

I used the absurdity of this notion to illustrate the inherent
absurdity in Godel's claims.

Obviously, it illustrates nothing of the kind.
Godel wasn't claiming or trying to "add a single sentence"
to the LANGUAGE, dumbass. It gets added to the AXIOM-set
for the THEORY.

Look at the outline above. The problems arise because what Godel
effectively did is to encode sentences of the form (*) into the
language of PA, If you want to claim that (*) has been added as an
*axiom* to PA as a result of Godel's translation procedure, that is
open to all kinds of objections. For one thing that would imply that
Godel carried out his translation procedure in a theory stronger than
PA, which would cross the boundaries of finitary reasoning. Secondly
axioms can be denied, and that would imply we could with equal
validity, deny Godel's incompleteness theorems.

Let me know what you think about this.

Regards, RS
.

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