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

On 4/24/2024 4:49 AM, Mikko wrote:

On 2024-04-23 14:54:09 +0000, olcott said:
 

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

On 2024-04-21 14:34:44 +0000, olcott said:
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On 4/21/2024 2:50 AM, Mikko wrote:

On 2024-04-20 16:37:27 +0000, olcott said:
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On 4/20/2024 2:41 AM, Mikko wrote:

On 2024-04-19 02:25:48 +0000, olcott said:
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On 4/18/2024 8:58 PM, Richard Damon wrote:

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

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*Mendelson (and everyone that knows these things) disagrees*
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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

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

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That does not follow. Besides, a reference to the term "undecidability"
is not a reference to the concept 'undecidability'.
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In other words you deny the identity principle thus X=X is false.

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It is not a good idea to lie where the truth can be seen.
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It is not a good idea to say gibberish nonsense and
expect it to be understood.
 >>> a reference to the term "undecidability"
 >>> is not a reference to the concept 'undecidability'.

 That is how a sentence must be quoted. The proof that the quoted
sentence can be understood is that Richard Damon undesstood it.
 

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)

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

 

It usually means one cannot make up one's mind.
In math it means an epistemological antinomy expression
is not a proposition thus a type mismatch error for every
bivalent system of logic.

 No, it doesn't. There is no reference to an epistemological
anitnomy in "undecidable".
 

...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)

not-⊢K ℬ and not-⊢K ¬ℬ. (Mendelson: 2015:208)
K ⊬ ℬ and K ⊬ ¬ℬ. // switching notational conventions
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Incomplete(F) ≡ ∃x ∈ L ((L ⊬  x) ∧ (L ⊬ ¬x))

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So not the same.

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When an expression cannot be proved or refuted is a formal system
this is exactly the same as an expression cannot be proved or refuted
in a formal system.

 To say about an expression that neither it nor its negation cannot be
proven is not the same as to say about a formal system that it contains
expressions that can neither be proven or disproven.
 

 > To say about an expression that neither it nor its negation cannot be
 > proven
means that it cannot be proven in some formal system
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer