Re: Undecidability based on epistemological antinomies V2 --Mendelson--

On 4/23/24 10:44 AM, olcott wrote:

On 4/22/2024 5:54 PM, Richard Damon wrote:

On 4/22/24 10:03 AM, olcott wrote:

On 4/22/2024 3:26 AM, Mikko wrote:

On 2024-04-21 14:34:44 +0000, olcott said:
>

On 4/21/2024 2:50 AM, Mikko wrote:

On 2024-04-20 16:37:27 +0000, olcott said:
>

On 4/20/2024 2:41 AM, Mikko wrote:

On 2024-04-19 02:25:48 +0000, olcott said:
>

On 4/18/2024 8:58 PM, Richard Damon wrote:

>

Godel's proof you are quoting from had NOTHING to do with undecidability,

>
*Mendelson (and everyone that knows these things) disagrees*
>
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf

>
On questions whether Gödel said something or not the sumpreme authority
is not Mendelson but Gödel.
>

>
When some authors affirm that undecidability and incompleteness
are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability.

>
That does not follow. Besides, a reference to the term "undecidability"
is not a reference to the concept 'undecidability'.
>

>
In other words you deny the identity principle thus X=X is false.

>
It is not a good idea to lie where the truth can be seen.
>

>
 >>>"undecidability" is not a reference to the concept 'undecidability'.
That is the best that I could make about the above quote. There is no
standard practice of using different kind of quotes that I am aware of.

>
Except that undeciability and incompleteness are not the EXACT same thing.
>

 So you were paying attention?
He said that undecidability is not the same thing as undecidability.
Somehow he felt that two different kinds of quotes mean something.

No, he was talking about how the CONCEPT of undecidablity is not the same as the word. (cf, "This is not a pipe")
Godel was NOT "talking about undecidability" but "incompleteness" as that was the term he was using, and trying to replace words with words that they are closecly related to was the error that

 

They CAN'T be, because they apply to different class of objects.
>
Of course, you are too stupid to understand that, because you logic is based on making category errors.

 In this case the issue is that you did not pay attention.
You glanced at a couple of words without even seeing them
and then spouted off a canned rebuttal that does not apply.

 

>

An undecidable sentence of a theory K is a closed wf ℬ of K such that
neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and
not-⊢K ¬ℬ. (Mendelson: 2015:208)

>
So that is what "undecideble" means in Mendelson: 2015. Elsewhere it may
mean something else.
>

>
It never means anything else.

>
LIE.
>
It also means (as the ORIGINAL definition) a computation problem for which no computation can be created that always gives the correct answer.
>

 That is the theory of computation way of saying it.
Mendelson translates the same idea into the math way of saying it.

As far as I have seen, Math, didn't start with the term undecidable, but adopted it when the concept that a proof and a program could have a correspondance.

 

>

Incomplete(F) ≡ ∃x ∈ L ((L ⊬  x) ∧ (L ⊬ ¬x))

>
So not the same.
>

>
Not provable or refutable in a formal system is exactly
the same as not provable of refutable in a formal system.
I think that you are playing head games.
>

>
But that isn't what the above says, itr says that F HAS a statement that is not provable or refutable, while undecidable (when applied to a statement) says THAT STATEMENT is not provable or refutable.
>
SYSTEMS are not STATEMENTS, so you are shows to be just wrong.

 When an expression is neither provable or refutable because it is not
a statement/proposition that has a truth value then it must be rejected as a type mismatch error for ever bivalent system of logic.

Right, proper logic system do not have as elements of their language statements that are not truth bearers.
Some how you seem to say that but also reject it, as you somehow think they are.
Your problem is that you don't seem to understand that there ARE, as PROVEN, statments that can be show to MUST be a truth bearer, and in fact can be shown in a meta-theory to be true in the theory, that can not be proven in the theory.
I think a big part of your problem is you don't understand what FORMAL LOGIC is and how Systems and Meta-Systems work.
Of course, since this has been pointed out to you, that just make all your proclaimations about them just pathological lies and proves your utter ignorance of the subject.

 A proposition is a central concept in the philosophy of language,
semantics, logic, and related fields, often characterized as the primary
bearer of truth or falsity.  https://en.wikipedia.org/wiki/Proposition