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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic,comp.theory
Subject: Re: Undecidability based on epistemological antinomies V2
 --Mendelson--
Date: Fri, 26 Apr 2024 12:15:40 -0500
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On 4/26/2024 11:38 AM, Ross Finlayson wrote:
> On 04/26/2024 08:28 AM, olcott wrote:
>> On 4/26/2024 3:42 AM, Mikko wrote:
>>> On 2024-04-25 14:27:23 +0000, olcott said:
>>>
>>>> On 4/25/2024 3:26 AM, Mikko wrote:
>>>>> epistemological antinomy
>>>>
>>>> It <is> part of the current (thus incorrect) definition
>>>> of undecidability because expressions of language that
>>>> are neither true nor false (epistemological antinomies)
>>>> do prove undecidability even though these expressions
>>>> are not truth bearers thus not propositions.
>>>
>>> That a definition is current does not mean that is incorrect.
>>>
>>
>> ...14 Every epistemological antinomy can likewise be used for a similar
>> undecidability proof...(Gödel 1931:43-44)
>>
>>> An epistemological antinomy can only be an undecidable sentence
>>> if it can be a sentence. What epistemological antinomies you
>>> can find that can be expressed in, say, first order goup theory
>>> or first order arithmetic or first order set tehory?
>>>
>>
>> It only matters that they can be expressed in some formal system.
>> If they cannot be expressed in any formal system then Gödel is
>> wrong for a different reason.
>>
>> Minimal Type Theory (YACC BNF)
>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>
>>
>> I created MTT so that self-reference could be correctly represented
>> it is conventional to represent self-reference incorrectly. MTT uses
>> adapted FOL to express arbitrary orders of logic. When MTT expressions
>> are translated into directed graphs a cycle in the graph proves that
>> the expression is erroneous.
>>
>> Here is the Liar Paradox in MTT: LP := ~True(LP)
>> 00 root (1)
>> 01 ~    (2)
>> 02 True (0) // cycle
>> Same as ~True(~True(~True(~True(...))))
>>
>> In Prolog
>> ?- LP = not(true(LP)).
>> LP = not(true(LP)).
>> ?- unify_with_occurs_check(LP, not(true(LP))).
>> false.
>> Indicates  ~True(~True(~True(~True(...))))
>>
>> In mathematical logic, a sentence (or closed formula)[1] of a predicate
>> logic is a Boolean-valued well-formed formula with no free variables. A
>> sentence can be viewed as expressing a proposition, something that must
>> be true or false.
>> https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>>
>> By definition epistemological antinomies cannot be true or false thus
>> cannot be logic sentences therefore Gödel is wrong.
>>
> 
> Actually what results is that Goedel refers to a particular kind
> of enforced, opinionated, retro-Russell ordinarity, that sees it
> so that "logical paradox" of quantifier ambiguity or quantifier
> impredicativity, resulting one of these one-way opinions, stipulations,
> assumptions, non-logical axioms of restriction of comprehension,
> makes it sort of like so for Goedel as "completeness, you know,
> yet, incompleteness, ...".
> 

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

epistemological antinomies cannot be true or false thus cannot
be propositions that must be true or false.

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)

Undecidable(K, ℬ) ≡ ∃ℬ ∈ K ((K ⊬ ℬ) ∧ (K ⊬ ¬ℬ))

To hazard a guess about what you mean, or to precisely state exactly
what I mean there is no such ℬ in K because such a ℬ in K could not
be a proposition of K.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer