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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic,comp.theory
Subject: Re: Undecidability based on epistemological antinomies V2
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Date: Fri, 26 Apr 2024 10:28:08 -0500
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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.

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