2018-10-26 17:03:12 UTC
A Theorem is essentially nothing more than an expression of language that can be proven True entirely on the basis on known facts, thus the conclusion of sound deductive inference.
More formally Theorems are mathematical proofs based on axioms and rules-of-inference only without specifying any premises, thus forming sound deductive inference.
Axioms can be thought of as expressions of language that have been expressly defined to have the semantic property of Boolean True.
Rules-of-inference show how to correctly transform some expressions of language into other expressions of language.
Any expression of language not having a Boolean Property with a value of True or False is not a correct Declarative sentence / sentence of logic.
This universal Truth predicate rejects as malformed any expressions of language that would otherwise show Incompleteness or Undefinability: ∀x ∈ F True(F, x) ↔ Theorem(F, x)
For all expressions x in language L, x is True in L if and only if x is a Theorem of L
(G) F ⊢ GF ↔ ¬ProvF(⌈GF⌉).
Simplified to this:
G ↔ G ∈ F ~Provable(F, G)
“This sentence is not provable”
G ≡ ~Provable(G) // To avoid clutter F is implied rather than stated
G asserts that G is not Provable. If G was a theorem (thus true) then it would contradict its assertion that G is not Provable.
~G asserts that G is Provable. If ~G was a theorem (thus true) then it would contradict its assertion that G is Provable.
∴ Since Neither G nor ~G is a theorem (true) G is rejected as an incorrect Boolean Proposition because it is neither True nor False.
Copyright 2018 by Pete Olcott