Re: The philosophy of logic reformulates existing ideas on a new basis --- infallibly correct

olcott <polcott333@gmail.com> wrote:

On 11/9/2024 12:47 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 11/9/2024 11:58 AM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 11/9/2024 10:03 AM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.

No.  Unprovable will remain.

*Like I said you don't pay f-cking attention*

Stop swearing.  I don't pay much attention to your provably false
utterances, no.  Life is too short.

That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.

Not at all.  I denigrate your lies, where by lies I mean the emphatic
utterances of falsehood due to a lack of expertise in the subject matter.
See the beginning of this subthread.

You are not doing that. I am redefining the foundation
of the notion of a formal system and calling this a
lie can have your house confiscated for defamation.

Hahahaha!  You lack the expertise to redefine formal systems.  What
you'll end up with, if anything at all, will be an incoherent,
inconsistent mess.  My house, should such exist, is in no danger.

You are the one with reckless disregard for the truth.  You haven't even
bothered to read the introductory texts which would help you understand
what the truth is.

[ Spam removed ]

Hint: Gödel's theorem applies in any sufficiently powerful logical
system, and the bar for "sufficiently powerful" is not high.

Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).

If you're going to redefine the word provable to mean something else,
you'll need some other word to mean what provable means to everybody
else.

I am correcting the somewhat ill-founded notion of provable
to only mean applying truth preserving operations to finite
string expressions of language.

[ Spam removed ]

That's what everybody else means by provable, too.  That's what Gödel
meant when he wrote the paper with his famous theorem in it.

Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)

If you're going to change the standard meaning of standard words, you'll
find communicating with other people somewhat strained and difficult..

ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.

No, they didn't do the same thing.  They stayed within the bounds of
logic.

ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORY

[ Spam removed ]

No.  Naive set theory had proven itself to be incomplete or inconsistent.
It needed changing or replacing.

 And yes, they resolved a paradox.  There is no paradox for your
"system" to resolve, even if it were logically coherent.

When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.

OK, That's a proof by contradiction that ~provable cannot mean ~true. 

The assumption that ~Provable(PA, g) does not mean ~True(PA, g)
cannot correctly be the basis for any proof because it is only
an assumption.

It is an assumption which swifly leads to a contradiction, therefore must
be false.  But you don't understand the concept of proof by
contradiction, and you lack the basic humility to accept what experts
say, so I don't expect this to sink in.

We know, by Gödel's Theorem that incompleteness does exist.  So the
initial proposition cannot hold, or it is in an inconsistent system.

Only on the basis of the assumption that
~Provable(PA, g) does not mean ~True(PA, g)

No, there is no such assumption.  There are definitions of provable and
of true, and Gödel proved that these cannot be identical.

Get rid of that single assumption AND EVERYTHING CHANGES

It's not an assumption, it's a proven fact, much like 2 + 2 = 4 is a
proven fact.

[ .... ]

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

--
Alan Mackenzie (Nuremberg, Germany).