Re: Help Clarify Godel sentence

herbzet wrote:

Nam Nguyen wrote:
Peter_Smith wrote:
On Sep 7, 7:51 am, Nam Nguyen <namducngu...@xxxxxxx> wrote:
All what your long sentence here really says is G and G' are significantly
because they're different! (Note I never stated they are identical!).
You're of course free to call anything "significant" since the adjective
is subjective; you just simply failed to demonstrate why the difference
between 2 specific G's is significant to GIT1.
Ye gods.

Suppose T and T* are different theories. Apply the same Gödelian
construction to T and T* and the output Gödel sentences G and G* will
in general be different -- significantly different in not being
logically equivalent, yes?
No. AC and ~AC are not logically equivalent: which one is significant
and which one is not?

Since each is the denial of the other, they are surely significantly
different propositions under any interpretation.

"Any interpretation"? Really? Let T1 = {AC /\ ~AC}. Are AC and ~AC
2 "significantly different propositions"?


Of course, syntactically the difference is trivial -- one just has
a negation symbol prepended.

Since you don't seem to understand, let me
repeat you have to define what "significant" is; and to me being
non logically equivalent is a poor definition.

In the current context, it's a fine definition.

According to you right above, syntactical difference is "trivial".
Also, though AC and ~AC aren't logically equivalent, I've demonstrated
the difference between AC and ~AC is not significant w.r.t. T1.
So, I Peter and you got to refute why my demonstration is wrong.
Just hand waving "In the current context, it's a fine definition"
is not saying anything as far as refuting/debating is concerned!


In particular, if T* = T+G, then certainly G* is not logically
equivalently to G, yes?

Because T* entails G but not G*: so G is not a Gödel sentence for T*
but G* is, yes?
Didn't I say about what I defined as "significant":

>>> At least not in fundamental ways that the proof of GIT1 would depend on!

That is true, but utterly *irrelevant* to the current context. It
is you who have inserted the consideration of how the general proof
of GIT1 would or would not depend "fundamentally" on the particulars
of a given G(T) into the conversation.

It's OK for you to use the word "irrelevant". But you got to technically
*defend your statement*: just saying that is no way helping the discussion!
(Hint: GIT1's proof cant' depend on the syntactical difference of G and
G' or their being "non logically equivalent"!)


Of course all sentences G(T) are generically alike, regardless
of the specifics of T -- but that was not what was being discussed.

Unless you clearly articulate what you really meant by "generically alike",
your assertion that "but that was not what was being discussed" has no ground!

The question was, specifically, how G(T) differs from G(T + G(T)).
They do differ, syntactically and in logical strength. Your answer --
that they are "fundamentally" alike -- is no answer at all to the
question posed.

Your "T* entails G but not G*" is a *consequence* of GIT1, how could that
be significant to the the proof of GIT1.

No one asserted that it was. You are engaged in an imaginary dispute.

It's not an imaginary dispute. You should reflect on the conversation between
Peter and I had, surrounding Peter's response to me to understand that
it's not:

Peter wrote:

> Your remark "if G and G* are to be different, it'd be for superficial
> purposes" suggested otherwise. UNless you think that's being and not-
> being a theorem of a given theory is a merely superficial difference!

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

Relevant Pages

  • Re: Help Clarify Godel sentence
    ... You're of course free to call anything "significant" since the adjective ... No. AC and ~AC are not logically equivalent: ... >>> At least not in fundamental ways that the proof of GIT1 would depend on! ... we must therefore examine the methods of the mathematician." ...
    (sci.logic)
  • Re: Help Clarify Godel sentence
    ... You're of course free to call anything "significant" since the adjective ... non logically equivalent is a poor definition. ... >>> At least not in fundamental ways that the proof of GIT1 would depend on! ... we must therefore examine the methods of the mathematician." ...
    (sci.logic)
  • Re: Help Clarify Godel sentence
    ... You're of course free to call anything "significant" since the adjective ... non logically equivalent is a poor definition. ... You say it's a poor definition; I say it's a fine definition. ... >>> At least not in fundamental ways that the proof of GIT1 would depend on! ...
    (sci.logic)
  • Re: Help Clarify Godel sentence
    ... non logically equivalent is a poor definition. ... and I could defend my reasons. ... ~AC are non-logical formulas, but their being non-logically-equivalent ... >>> At least not in fundamental ways that the proof of GIT1 would depend on! ...
    (sci.logic)
  • Re: Godel Contradiction
    ... Daryl McCullough wrote: ... but they can be logically equivalent (the left side ... we must therefore examine the methods of the mathematician." ... (Shoenfield, "Mathematical Logic") ...
    (sci.logic)