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

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.
Bivalent formal systems of logic only operate on propositions
thus any expression that is not a proposition is a type mismatch error.
A proposition is a central concept in the philosophy of language,
semantics, logic, and related fields, often characterized as the
primary bearer of truth or falsity. https://en.wikipedia.org/wiki/Proposition
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)
AKA Undecidable(K, ℬ) ≡ ∃ℬ ∈ K ((K ⊬ ℬ) ∧ (K ⊬ ℬ))
Mendelson, Elliott 2015. Introduction to Mathematical Logic sixth
edition CRC Press Taylor & Francis Group Boca Raton, FL
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