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

Am Sat, 09 Nov 2024 13:00:22 -0600 schrieb olcott:

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:

On 11/9/2024 5:01 AM, joes wrote:

On 11/8/24 12:25 PM, olcott wrote:

 

That formal systems that only apply truth preserving
operations to expressions of their formal language that
have been stipulated to be true cannot possibly be
undecidable is proven to be over-your-head on the basis
that you have no actual reasoning as a rebuttal.

Gödel showed otherwise.

That is counter-factual within my precise specification.

That's untrue - you don't have a precise specification.  And even
if you did, Gödel's theorem would still hold.

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.

Go on, sue him, liar.

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.
I have no fear of you starting a defamation case against me.  For a
start, you'd have to learn some German, and for another thing, I'd win
on the merits.
 

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).

That doesn’t make ~g provable.

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.

What else do you think it meant?

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
 

 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’s a very safe assumption, as it keeps both possibilities for the
truth value of g open.

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)
Get rid of that single assumption AND EVERYTHING CHANGES

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.