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

On Mar 19, 12:09 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:
On Mar 18, 10:24 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:



On Mar 17, 4:40 pm, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

On Mar 17, 10:34 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:> On Mar 15, 10:18 am, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:

After all,
there is no way for a powerless individual like me to change human
nature. But sooner or later, the farcical nature of this state of
affairs will become clear to everybody and hopefully some one of a
suitable stature and integrity will break away from the status quo and
seriously consider my contribution.

Yes, you are a great revolutionary thinker, and not until some other
great revolutionary thinker who also happens to have a research grant
comes along will mankind ever be free from the cave of ignorance.

Why don't you just answer the point I have made?

Because whatever point you think you made there does not interest me,
especially as your argument for it is so confused.

It does not look confused to me. It is crystal-clear. I am going to
snip out most of the NAFL-related points in this post and only address
this issue in this thread. We will discuss NAFL later. Unwillingness
on the part of your peers to discuss NAFL is obvious, I don't need to
prove that. But since you are willing, I take that as a positive
sign.

Good. In the meantime, when you go off describing yourself as poor
widdle misunderstood leader of mankind from ignorance if only for an
official pat on tht back from a real live heavyweight mathematician,
then I'll feel free to ridicule that.

Again, observe the *behaviour* of your "heavyweight" peers, words are
worthless. A new logic like NAFL is self-evidently noteworthy and that
explains their silence. An attempt to ridicule NAFL would be like a
fool's venture.... and your "heavyweight" peers are not fools. I will
grant that much to them.

But this post makes a point that does not invoke
NAFL.

It invokes confusion.

Why can't I find a straightforward individual from amongst your
peers, someone who will engage me in a honest debate?

Debate what? One cannot debate nonsense.

The point made is very clear. I have attempted to explain it again in
this post.

What is the
point in denying credit to an individual for legitimate work that he
has done,

Who said you should be denied any credit for any work you're done?

Actions speak louder than words.Ask your peers. I bet they will not
reply to you either if you so much as mention NAFL.

even if one ignores the unethical and petty nature of such
an enterprise?

What unethical and petty enterprise are you referring to?

Self-evidentl, but let us not waste time on polemics here. See the
points made below and try to respond, if possible.

I can understand and rationalize the existence of a few
individuals (e.g. closer to home) who are of this nature, but the
academic world seems to have mass-produced clones of this breed.

Right. Because people are not impressed with your work, it must be
that they are mass-produced clones. God forbid you countenance that
each individual is not impressed with your work based on that
individual having come to that conclusion himself.

You don't understand my work. I do. How can you be either impressed or
not impressed? As for your peers, they have said nothing so far (at
least not in public). So how do you draw that conclusion? People
apparently were not impressed with non-Euclidean geometry either for
decades, until the originators died without getting due credit. Petty
behaviour in the extreme, to deny due credit to people now recgonized
as brilliant.

The "true reason" for incompleteness is that Godel overreached beyond
the boundaries of first-order logic.

What are you talking about? Do you contend that the proof is not
formalizable in PRA or Robinsion arithmetic or PA or even Z set theory
- all first order theories.

The proof is formalizable with "coding". I have clearly said that I am
questioning the legitimacy of the coding and you have failed to
understand that. You need to remove the blinkers and take another
objective look at what I have said. I have said it again below.

You might ask, what is the
objection to second-order logic, why not formally admit propositions
which quantify over "all" propositions?

Who objected to second order logic?

I did. What I stated was my objection.





Take the assertion that "From
a contradiction, an arbitrary propostion follows", or , say,

P&~P --> Q,    (*)

Let us take P to be fixed in (*), and consider this propositon for
arbitrary Q. Obviously there is an implied quantification over Q,

If 'Q' is a sentence letter, then it is not quantified over in the
language. (Of course, the meta-language may quantify over sentence
letters of the object language).

If 'Q' is a meta-variable in the meta-language, then it is not
quantified over in the meta-language (if the meta-language is first
order).

I object to this assertion that there can be a variable in a language
that is not quantified over. All variables range over some domain and
that *is* quantification. This is the kind of confused thinking that
you are loudly accusing me of. EIther Q is a constant (a fixed
propostion, which must be specified by construction) or else it is
quantified over. If Q is a variable in the metalanguage, it is
quantified over in the metalanguage. E..g when you say that P&P->Q in
the metalanguage, you are also asserting that this proposition holds
for an arbitrary P and arbitrary Q. That is quantification. The point
is that the object (first-order) language does not contain any such
variables because actually propostions in the first-order language are
obtained after substitution of P and Q by specific formulas.

if
we want to convey that "all" propositions follow from a contradiction.
If we admit (*) as a formal proposition in the same language which we
are considering the notion of "all" propositions (e.g. PA), then
clearly (*) is an impredicative construction,
for the quantification
must include (*) as well in its domain.

IF we admit it thus then it's a problem. But we DON'T. The assertion
that all sentences follow from a contradiction is not made in the
object language but rather in the meta-language.

Ah, so you do agree that there is quantification in the metalanguage.
Fine. Let us take it from there.

This is not permitted in NAFL,
which requires propositions like (*) to be metamathenmatical
constructions, outside the formal language. Then you can see that
there is no impredicativity.

(1) You told us before that the syntax of NAFL is that of classical
first order logic. (2) There's not even the problem you seem to think
in classical first order logic, since we DON'T conflate the meta-
language and the object language.

Of course, you say you don't. But with Godel's coding, I am alleging
that you are doing that tacitly and objectionably. And I have stated
my objection precisely. Here it is again.

Godel translated a proposition of the form P&~P->Q into a proposition
in the *object* language of arithmetic, say, PA. Call this proposition
S. Note that S is a specific propositon involving only numbers.

By your own assertion above you do not admit P&~P->Q in the object
language of arithmetic. So if you scan all proofs of PA, you will not
find a proof of P&~P-->Q (a proof must end with the propositon
proven). You will not find a refutation of P&~P --> Q either in the
list of PA-proofs. For if you did, then P&~P&~Q must hold in every
model of PA, for some specific (number-theoretic) propostions P and Q
and we know that is not possible.

Now Godel translated the meta-theoretical propostion P&~P->Q into a
specific number-theoretic propostion S in the language of PA. Note
that S is not a variable; it is a constant and we have a construction
for S. Since we just now argued that P&~P-->Q is not either provable
or refutable in PA, it follows that S must also be undecidable in PA.
But this means that there must exist a model of PA in which S is
false, and such a model cannot exist, for the same reason argued
earlier. This is a contradiction.

In fact my understanding is that PA proves S. If the meta-theoretical
sentence P&~P->Q does translate to S and if this coding is accepted as
legitimate, this is tantamount to the claim that PA proves P&~P-->Q,
which contradicts our earlier assertion that this is not possible. So
now you must maintain that the metatheoretical proposition P&~P-->Q
becomes a formal sentence of PA via coding and *is* provable in PA,
after all. This is precisely what I am objecting to as stepping beyond
the boundaries of first-order logic.

If S were undecidable in PA and if you could justify the
undecidability of P&~P->Q in PA, then I would not have this objection.
For one could assert that S has a dual metamathematical interpretation
as the sentence P&~P->Q which we know is neither provable nor
refutable in PA and as confirmed by the supposed undecidability of S.
But unfortunately S *cannot* be undecidable in PA (by the argument
given earlier) and is in fact not undecidable in PA. This amounts to a
serious objection to the kind of translation procedure employed by
Godel.

Kindly reply to ths post. I am happy that you are attempting to take
the bull by the horns and I would welcome your critical scrutiny of
NAFL too. I will make a start this week-end (although I must take
office work also into my holiday, so my responses will not be
immediate).


To put my objection to Godel's coding in a nutshell. By the metatheory
in which first-order languages are constructed, P&~P->Q is not a
propostion in the object language of PA and therefore cannot be
provable in PA. It cannot be refutable in PA as argued earlier. Since
it is not a formal propositon in any first-order language,, P&~P--> Q,
we are barred from invoking model theory (via the completeness
theorem) and drawing the (unmaintainable) conclusion that there must
exist a model of PA in which P&~P-->Q is false. That is how we escape
from the contradiction that would arise of P&~P->Q were to be admitted
into the object language and were to be formally undecidable in PA.

Enter Godel. He translated P&~P->Q into an arithmetical proposition S
in the object language of PA. If we maintain that S still has only a
metatheoretical interpretation as P&~P-->Q, then we must argue that S
has to be undecidable in PA, so that we may avoid contradicting the
earlier metatheoretical argument for P&~P-->Q being neither provable
nor refutable in PA. In that case we can no longer avoid the model-
theoretic conclusion (from the completeness theorem) that there must
exist a model for PA in which S is false, for S is indeed a legitimate
sentence in the first-order language of PA. But if we hold that S does
indeed translate to P&~P->Q, such a model cannot exist. We are
therefore forced to rule out undecidability of S in PA.

In fact S has to be provable in PA, and by Godels translation
procedure, the sentence "PA |- S" is also provable in PA. Clearly we
now have a metatheory in which P&~P->Q is provable in PA. This
contradicts our earlier metatheory in which P&~P--> Q is not provable
in PA. Which of these metatheories is acceptable? I reject the coding
used by Godel as amounting to infinitary reasoning that crosses the
boundaries of first-order logic and leads to other infinitary
consequences, e.g. the existence of nonstandard models of PA.

Regards, RS

.

Relevant Pages

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