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

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:
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On 4/25/2024 3:26 AM, Mikko wrote:

epistemological antinomy

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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.

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That a definition is current does not mean that is incorrect.
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...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)
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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?
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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.
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Minimal Type Theory (YACC BNF)
https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
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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.
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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(...))))
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In Prolog
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
Indicates  ~True(~True(~True(~True(...))))
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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)
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By definition epistemological antinomies cannot be true or false thus
cannot be logic sentences therefore Gödel is wrong.
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 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