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

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.

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.

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

So not the same.
--
Mikko