Re: My talk about Godel to the post-grads.

MoeBlee wrote:
On Jul 15, 11:24 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
MoeBlee wrote:
On Jul 9, 6:36 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
Chris Menzel wrote:
You seem to be under the same sort of misimpression as JJ, viz., that
the content of Gödel's technical results is somehow a matter of
*opinion*; that their mathematical interpretation is open to debate.
Why is it not a matter of opinion when:
(a) This is not first order theorem where the syntactical Rules of Inference
and axioms are *well defined*.
Speaking of your misunderstandings. You've been told over and over and
over that the proofs of the incompleteness theorems can be carried out
in FORMAL first order theories such as Z set theory (and even weaker).
How do you know this 1st order formal system Z is consistent?

For godsakes, the question wasn't whether Z is consistent but rather
whether the incompleteness theorems are PROVABLE in Z. Please! Have
the honesty to recognize that your claim was incorrect that the
incompleteness theorems are not provable in a formal first order
theory and don't try to evade that by taking a DIFFERENT question of
whether Z is consistent.

Then, why don't we take a *much simpler* theory Z' = {GC + ~GC}? Not only
Z' proves the incompleteness theorems, it will be strong enough to prove
any most difficult mathematical problems in human history!

Really, Moeblee, since when could foundational issues of FOL reasoning be
independent of theory consistency? (Don't you think that G(T) which is about
incompleteness, and CON(T) which is about consistency have some relationship?)

MOREOVER, as I have read, as is widely recognized (though I haven't
personally confirmed the details), incompleteness is provable in a
system as weak as PRA (and, if I'm not mistaken, using only
constructive reasoning). Sure, you could even doubt the consistency of
constructive PRA (if 'constructive PRA' is not even a redundancy?),
but then what's the point? For then you might as well doubt the
consistency of ANYTHING. I mean, if you doubt the consistency of
constructive reasoing about finitistic mathematical events, then I
can't imagine anything you WOULDN"T doubt. And if you doubt everything
mathematical, then what is the point of singling out the
incompleteness theorems?

Give me axioms and rules of inference, then I'll believe theories such
as {Ax[x=a]}}, {Axy[x+y=0]}, ... are consistent. So, you now can correct
your imagination: there are things - theory consistencies - I do *not*
doubt!

Oh, but I recall now. When we spoke about this a while back, indeed,
you wouldn't even assent to the most primitive reasoning about short,
finite strings of symbols. So, of course, with doubts as pervasive as
that, nothing could satisfy you.

Again, you're wrong: there are theory consistencies that I'm satisfied.
(Hint: perhaps you'd want to minimize the number of times you're wrong,
by refraining from asserting what *others* might know, think, doubt, ...!)

You can verify that for YOURSELF just by following along each step in
textbook proofs. It is just willful ignorance on your part to keep
insisting that the incompleteness theorems are not provable from a
first order formal axiomatization and formal rules of inference.

What then is a meta theorem, to you?

I use the term 'meta-theorem' just as it is ordinarily used in
mathematical logic. There's no need for me to perform a ritualistic
definition for you, especially since I don't use it as TECHNICAL term.

That's the 1st clue here of your problems: 'meta-theorem' is a technical
term, and should be treated as such!

Specifically, a meta theorem is an *assertion* *about* formal reasoning of
a reasoning framework (e.g. FOL, SOL, etc...). Where as, e.g. in FOL, axioms
are the originating points of a proof, provability of a meta theorem flows
from *knowledge* *about* the formal reasoning framework in question.

Examples of true meta theorems (or just meta theorems):

- A FOL formula is finite in length.
- GC is not an axiom of Q (as defined by, say, Shoenfield)
- If ~GC is provable in Q then it's provable in any extension of Q.
...

Examples of false meta theorems (or just not meta theorems):

- A FOL proof is not finite in length.
- GC is an axiom of Q (as defined by, say, Shoenfield)
- If Q is a consistent system, both GC and cGC are provable.
...


Can you give an example of a meta theorem that's *not* a
first order theorem?

Any theorem, whether a meta-theorem or not, that is stated in second
order language is not a first order theorem, though, of course, in
many cases, there may be a first order version too.

That's the 2nd clue here of your problems: I asked you for *a meta-theorem*,
and you came back with an answer that's twice nonsensical:

- "Any theorem, *whether a meta-theorem or not*"!
- Basically what you said is a 2nd order theorem is not a 1st order theorem!
How does that become an example of a meta theorem?

You got to understand a question clearly, and answer straight to the point.
Of course you could have said "I don't know what a meta theorem is". But you
didn't.


Anyway, what is the point of these questions?

Because the edifice of proof is different between a FOL theorem and
a meta theorem. And one has to understand the difference to appreciate why
GIT is a meta theorem. Naturally.

The fact that the
incompleteness theorems are provable in many ordinary formal first
order theories is not affected by my definition of 'meta-theorem' or
by the fact that there are also second order and higher order meta-
theorems too.

Sure. Z' above is an ordinary formal first theory in which incompleteness
theorems are provable. Which of course is true. But which is the only
thing you could say about? How about the inconsistency of Z? Oh I forgot:
according to you, the issue here wouldn't be about inconsistency, it would
be *only* about provability of incompleteness theorems in Z'!


In GIT, what would be the
axioms for the condition "T be consistent", if GIT were a 1st order
theorem? You don't seem to know what you are talking about.

I TOLD you a THOUSAND times, the language of Z set theory is
sufficient to define 'is consistent' and the axioms of Z set theory
sufficient to prove the incompleteness theorems. Why or why or why
don't you understand that?

Because *you* have *not* demonstrated that *syntactically* a formula
of the form F /\ ~F can't be a theorem of Z!


What don't you understand when an author of a book in mathematical
logic says at the beginning that in principle all of the results in
the book can be formalized in a formal set theory?

In the name of logical arguing, what is "in principle all of the results"
doing here? You meant "in practice" some of the results can't be formalized?

Moreover, at what
point in reading a book of mathematical logic did you ever notice a
definition or proof that could not be carried out in a formal set
theory?

At any point! Because GIT is a *meta* theorem, *not* a FOL theorem!


(b) We don't know which formal system would 100% characterize the naturals.
Whatever "characterize" means (I haven't yet read all the subsequent
discussion), I don't know why you think that has any bearing on the
fact that the incompleteness theorems are provable in a formal first
order theory, which has an associated formal semantics.

Quit being ridiculous! If you "haven't yet read all the subsequent
discussion" then of course you "don't know why [i] think ....".
Go and read the other posts first! Before responding again!

YOU don't be ridiculous. It is common posting practice to respond to
posts down the line without necessarily having read all the posts
subsequent to the post one is responding to. That is not ideal, but
usually it is just impractical to have to read an entire thread before
going back to respond to a post somehere earlier in the the thread.

Sure. It's impractical. So stop saying thing like "I don't know why you
think that...". When you don't understand something the other guy has
just said and the other posts and you don't want to read them, Oh well
what can anybody do to help you?


Again, whatever certainty you require, the incompleteness theorems are
NO LESS formally provable than many another ordinary mathematical
theorem.
That's right there's no meta theorem anymore!

What a strange non sequitur. The fact that certain meta-theorems, such
as the ordinary incompleteness theorems, are provable in a first order
theory doesn't mean that they aren't meta-theorems. Are you, by
chance, under the misconception that a meta-theory for a first order
theory can't itself be a first order theory?

What's the difference between a model of a meta "theory" and that of
a formal theory? What does it mean for a meta theory to be inconsistent?


Or is 0+0=0 a meta theorem in, say, Q too?
Again, what a strange question.

Again, you refused to answer a very short technical question in a discussion!
That's not a good sign about your understanding of the foundation issues
being debated!

So it is inexplicable why you fret about proof of the incompleteness
theorem as opposed to proof of many another ordinary mathematical
theorem.
I already did in the other posts: GC, cGC, encoding primes, etc...
You either didn't read it or simply don't understand it!

I don't claim to have read every post SUBSEQUENT to the post to which
I am responding nor to base my responses to a post with consideration
of all SUBSEQUENT posts. This is called 'cross-posting' and I find
that usually reasonable posters understand this aspect of the medium
and allow each other latitude in consideration of the practical
inevitability of cross-posting, as it would be quite unmanagable for a
poster to have to first read all subsequent posts before going back to
an earlier post to respond to it.

There's no "cross posting", as you imagined! You got into a conversation
of 2 other people in *one* thread and accused one of them all sort of things
about what he had said to the other. It's your responsibility to understand
clearly what they had argued about, *before* making any accusation! If you
can't do that, then don't debate with any of them!

What would have happened if, say, Hilbert had said to Godel: "Herr Godel,
you don't have an opinion or choice here, you have to use *only* syntactical
means through axioms and Rules of Inference, to prove G(T) is undecidable"?
Then Godel would have said, "It's easy to see that my proof is
formalizable in a system of axioms and rules of inference."
Goedel, being a genius, would never say that, you know! When he said
"is true but not provable" for sure there's a technical reason why
he couldn't have instead said "is syntactically undecidable".
Do you understand that?

I didn't suggest anything about the phrase "syntactically undecidable".

What's your point? Obviously I suggested the phrase, not you!

I simply point out that the incompleteness theorems can
be proven in a formal first order theory and that there's no reason to
think Godel didn't understand that, especially in the hypothetical
situation you raised (especially if it were at a later time when the
notion of a 'formal theory', in the sense of a recursively axiomatized
one, was widely understood).

I don't see how any of this answer any question. Let me put if differently.
Suppose I come to you and ask you to prove GIT using *only* syntactical
means through axioms and Rules of Inference, can you do it? You don't have
to let me know all the details, just a brief mentioning of the *major*
strategies how that could be done, that would be great. It could also be
great to if you say "I don't know how to".

So which would be the case, Moeblee?

Notice, by the way that Hilbert did NOT say that to Godel. And why do
you think that is? You don't think Hilbert too could see for himself
that Godel's proof is easily formalizable?
Why then we now don't have HIT, instead GIT?

Um, because Godel is the author of the proof?

If "Godel's proof is *easily* formalizable", don't you think a bright mathematician/
logician of Hilbert's level would have done that, way before Godel had some
insight about it?


MoeBlee




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