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

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. But what did
interest me is the passage I quoted.

Earlier your
objection was that you don't know what NAFL means.

You keep talking about "NAFL theories" and I keep asking for a
defintion of 'T is an NAFL theory'.

I will attempt to
address that soon.

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.

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.

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?

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

What unethical and petty enterprise are you referring to?

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.

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.

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?

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).

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.

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.

In fact there is no such thing as a tautology in NAFL, and (*) is not
a meta-theorem of NAFL theories. I will elaborate this week-end, if I
manage to find some time (I am on leave Thursday-Sunday, so hopefully
I will be able to get started on the long-pending NAFL thread).

Only you can say what goes on in NAFL. Since you've not defined what
an "NAFL theory" is, no one can by himself sit down to reason what is
or is not the case about NAFL theories. Our only hope is to wait each
time for you to make a pronouncement as to what is or is not the case
about NAFL theories and as we may not question your pronouncements
since they are not subject to independent evaluation upon any exact
defintion of what an NAFL theory is.

MoeBlee

.

Relevant Pages

  • Re: Godels comments about the "true reason" for incompleteness
    ... the farcical nature of this state of ... objection was that you don't know what NAFL means. ... why not formally admit propositions ... clearly is an impredicative construction, for the quantification ...
    (sci.logic)
  • Re: Non standard models of PA
    ... NAFL model is a different animal altogether from the classical model. ... it is all in the human mind as far as NAFL is ... NOT in the language of T itself. ... In NAFL undecidability is always with respect to NAFL theories. ...
    (sci.logic)
  • Re: Continuum hypothesis
    ... The first think you said in your alleged "discussion of NAFL" ... theorems" in the proof syntax of an NAFL theory. ... You CANNOT do that BECAUSE the CLASSICAL inference rules ... theories containing "all their legitimate propositions". ...
    (sci.logic)
  • Re: FOL/Intuitionistic logic versus NAFL. Part 1. Failure of non-contradiction
    ... be having at least 1% interest in NAFL. ... I will discuss the language and rules of inference of NAFL theories ... There are no absolute truths for formal propositions in the ... the human mind could choose different theories for T* (i.e., ...
    (sci.logic)
  • FOL/Intuitionistic logic versus NAFL. Part 1. Failure of non-contradiction
    ... be having at least 1% interest in NAFL. ... I will discuss the language and rules of inference of NAFL theories ... There are no absolute truths for formal propositions in the ... the human mind could choose different theories for T* (i.e., ...
    (sci.logic)