Re: Sorry Godel - All Truths are Provable


Rupert wrote:
Charlie-Boo wrote:
Rupert wrote:
Charlie-Boo wrote:
Daryl McCullough wrote:
Charlie-Boo says...

Let w be any true sentence and v be any provable sentence. Now |-v
since v is provable, and |-(w^v) since w^v is logically equivalent to v
since w is true.

Two statements A and B are *logically* equivalent only if the statement

A <-> B

is provable. So w^v is not logically equivalent to v unless

w^v <-> v

is provable.

Whatever! So do you agree that w^v is true iff v is true? Then what
do you call that? Anyway, the same reasoning applies. If |-P then
|-~~P and if |-P then |-P^P etc. Get it? So if two wffs have the same
truth value, then they obviously would have the same provability.

No, that's not obvious at all. It's not true.

It is obvious and not true (as it was obvious to Hilbert.)

Well, it's not obvious to me. Why is it obvious?

For the same reason it was obvious to Hilbert.

The idea that truth and provability are defined so differently because
of the way models force themselves in prematurely stifles the idea that
truth and provability are the same and we should look for where the
difference first occurs.

Call a sentence P r.e. iff its truth value equals its provability,
denoted by Px, i.e. it is true and provable, or it is false and not
provable.

Then if we can prove TRUE i.e. if |-TRUE then (|-TRUE)==TRUE so TRUE is
r.e., i.e. TRUEx. Thus Peano's axioms are equivalent to the
provability of TRUE.

So we begin with an r.e. wff TRUE. And then we can combine it with
other wffs. Are they r.e.? Is the result r.e.? Where do we reach a
wff that is not r.e.? These are the real questions.

C-B

That still
does not explain where logic fails us. Where is the "point of
disconnect"?

We all know from the outside that true does not mean provable (I have
developed at least a half dozen proofs of that myself.) But if your
car doesn't start, don't you look under the hood to see where the
problem is?

C-B

Why
not?

C-B

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Sorry Godel - All Truths are Provable
    ... So w^v is not logically equivalent to v unless ... So if two wffs have the same ... The idea that truth and provability are defined so differently because ...
    (sci.logic)
  • Re: Sorry Godel - All Truths are Provable
    ... Rupert wrote: ... Let w be any true sentence and v be any provable sentence. ... So w^v is not logically equivalent to v unless ... then they obviously would have the same provability. ...
    (sci.logic)
  • Re: Sorry Godel - All Truths are Provable
    ... Charlie-Boo wrote: ... Let w be any true sentence and v be any provable sentence. ... So w^v is not logically equivalent to v unless ... then they obviously would have the same provability. ...
    (sci.logic)
  • Re: Sorry Godel - All Truths are Provable
    ... Charlie-Boo wrote: ... Let w be any true sentence and v be any provable sentence. ... So w^v is not logically equivalent to v unless ... then they obviously would have the same provability. ...
    (sci.logic)
  • Re: The Law of the Excluded Middle again (long)
    ... "Inside the scope of quantification, ... to equate (or confuse) truth with provability." ...
    (sci.math)
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