previously undecidable logic sentences become decidable:

**When Closed WFF x of formal system F is considered:**

**True**----------its a theorem of F: ---------------

**(F ⊢ x)**

**False**----------its negation is a theorem of F:

**(F ⊢ ¬x)**

**¬True**---------its not a theorem of F: ----------

**(F ⊬ x)**

**¬Boolean**-----its neither True nor False in F:

**(F⊬x ∧ F⊬¬x)**

When-so-ever any closed WFF contains a ¬Boolean term the whole

WFF evaluates to ¬True thus maintaining consistency within the

**{axioms of truth}**adaptation to the notion of a formal system.

**Eliminating Undecidability and Incompleteness in Formal Systems**

https://www.researchgate.net/publicatio ... al_Systems

**The above adaptations show how this logic sentence is ¬Boolean thus ¬True:**

∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))

**Making the following paragraph ¬Boolean, thus ¬True**

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.

(Raatikainen 2018)