a **Collection** is defined one or more things that have one or more properties in common. These operations from set theory are available: **{****⊆, ∈****} **

An **BaseFact** is an expression X of (natural or formal) language L that has been assigned the semantic property of True. (Similar to a math Axiom).

A **Collection** T of **BaseFacts** of language L forms the ultimate foundation of the notion of Truth in language L.

To verify that an expression X of language L is True or False only requires a syntactic logical consequence inference chain (formal proof) from one or more elements of T to X or ~X.

True(L, X) ↔ ∃Γ ⊆ BaseFact(L) Provable(Γ, X)

False(L, X) ↔ ∃Γ ⊆ BaseFact(L) Provable(Γ, ~X)

**Since an expression X of language L is not a statement of language L unless and until it is proven to be True or False in L, every statement of language L can be proved or disproved in L.**

**∀L ∈ Formal_Systems
∀X ∈ L
Statement(L, X) → ( Provable(L, X) ∨ Refutable(L, X) )**

**Stanford Encyclopedia of Philosophy Gödel’s Incompleteness Theorems**

The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, **there are statements of the language of F which can neither be proved nor disproved in F.**

Formalized as: **Statement(F, G) ∧ ~Provable(F, G) ∧ ~Refutable(F, G)**

“This sentence is not true”. // is not true

// HOL with self-reference semantics

**LP ≡ ~∃Γ ⊆ BaseFact(L) Provable(Γ, LP)**

LP is rejected as semantically incorrect on the basis that no formal proof exists from one or more elements of T to LP or ~LP.

Because LP is neither True nor False LP is semantically incorrect.

“This sentence is not provable”. // is not provable

// HOL with self-reference semantics

**G ≡ ~∃Γ Provable(Γ, G)**

G is rejected as semantically incorrect on the basis that no formal proof exists from one or more elements of T to G or ~G.

**Copyright 2017, 2018 Pete Olcott **