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

On 11/11/2024 4:26 AM, Mikko wrote:

On 2024-11-11 03:08:36 +0000, olcott said:
 

On 11/10/2024 3:52 AM, Mikko wrote:

On 2024-11-09 18:05:38 +0000, olcott said:
>

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.
>

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).
>

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

>
But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA
and "~Provable(PA,g)" means that there is not. These meanings are don't
involve your "True" in any way. You may define "True" as a synonym to
"Provable" but formal synonyms are not useful.

>
We can ALWAYS prove that any expression of language is true or
not on the basis of other expressions of language when we have a
coherent definition of True(L,x).

 Not relevant.

It <is> relevant in that it does refute the Tarski
Undefinability theorem that <is> isomorphic to incompleteness.

The meaning of "Provable(PA,g)" does not depend on
the definition of "True(L,x)". "Provable(PA,g)" is false because
there is no proof of g in PA. For the same reason "Provable(PA,~g)"
is false.
 

There is no proof of Tarski's x in his Theory only
because x is incoherent in his theory.
https://liarparadox.org/Tarski_275_276.pdf
    Let {T} be such a theory. Then the elementary
    statements which belong to {T} we shall call the
    elementary theorems of {T}; we also say that
    these elementary statements are true for {T}.
    Thus, given {T}, an elementary theorem is an
    elementary statement which is true.
    https://www.liarparadox.org/Haskell_Curry_45.pdf
Haskell Curry is referring to a set of expressions that are
stipulated to be true in T.
We define True(L, x) to mean x is a necessary consequence of
the Haskell Curry elementary theorems of L.
(Haskell_Curry_Elementary_Theorems(L) □ x) ≡ True(L, x)
x = "What time is it?"
True(English, x) == false
True(English, ~x) == false
∴ Not_a_Truth_Bearer(English, x)
Under math rules we would declare that English is incomplete
because neither x nor ~x is provable in English.

There are actually infinitely many sentences of PA that could be used
instead of g to show incompleteness but one is enoubh.
 

That Gödel relies on True(meta-math, g) to mean True(PA, g)
is a stupid mistake that enables Incomplete(PA) to exist.

 Gödel proved Provable(meta-math, "~Provable(PA,g) ∧ ~Provable(PA,g)").
 

That is the same thing as proving:
This sentence is not true: "This sentence is not true" is true.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer