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

olcott <polcott333@gmail.com> wrote:

On 11/13/2024 5:57 AM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

[ .... ]

Incomplete(L) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
When the above foundational definition ceases to exist then
Gödel's proof cannot prove incompleteness.

*You just don't understand this at its foundational level*

You make me laugh, sometimes (at you, not with you).

What on Earth do you mean by a definition "ceasing to exist"?  Do you
mean you shut your eyes and pretend you can't see it?

It is very easy if your weren't stuck in rebuttal mode
not giving a rat's ass for truth you would already know.

So, no answer.  Just an attempted diversion.  As you ought to be aware,
definitions don't "cease to exist" when you shut your eyes.

A set as a member of itself ceases to exist in ZFC, thus
making Russell's Paradox cease to exist in ZFC.

Incompleteness exists as a concept, whether you like it or not.  Gödel's
theorem is proven, whether you like it or not (evidently the latter).

When the definition of Incompleteness:
Incomplete(L) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
  becomes
¬TruthBearer(L,x) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))

Then meeting the criteria for incompleteness means something
else entirely and incompleteness can no longer be proven.

The definition of incompleteness is what it is, and you can't redefine it
out of existence.  If you steal the word and make it mean something else,
then there will be some other word to take the place of incompleteness.
Your system _will_ be incomplete if it's consistent, with a mininum level
of capability.

After 2000 years most of the greatest experts in the world
still believe that "This sentence is not true" is undecidable
rather than incorrect.

As for your attempts to pretend that unprovable statements are the
same as false statements,

I never said anything like that.

You did.

You are so stuck on rebuttal that you can't even keep track on the
exact words that I actually said.

I never said that ~True(L,x) == False(L,x). That is an egregious
error on your part. I have been saying the direct opposite of your
claim for years now. False(L, x) == True(L, ~x)

There cannot possibly be any expressions of language that
are true in L that are not determined to be true on the
basis of applying a sequence of truth preserving operations
in L to Haskell_Curry_Elementary_Theorems in L.

There can be, and there are.  See Gödel's theorem.

https://www.liarparadox.org/Haskell_Curry_45.pdf

Everything that is true on the basis of its meaning
expressed in language is shown to be true this exact
same way, within this same language.

Ignorant garbage.

Logicians take the prior work of other humans as inherently
infallible. Philosophers of logic examine alternative views
that may be more coherent.

Logicians rigorously check the work of their peers  Philosophers, on the
other hand, don't have to prove things.

Mark Twain got it right when he asked "How many legs
does a dog have if you call a tail a leg?".  To which the answer is
"Four: calling a tail a leg doesn't make it one.".

--
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).