Syntactic_Logical_Consequence (Γ proves A)

A formula A is a syntactic consequence within some formal system FS of a set Γ of formulas if there is a formal proof in FS of A from the set Γ:

**Γ ⊢ _{FS} A**

Translation to Minimal Type Theory notational conventions

**Γ ⊢ _{FS} A ≡ ( Γ ⊂ FS (Γ ⊢ A) )**

Named predicate (of predicate logic PL) impossible to prove and impossible to refute:** G( ~∃Γ ⊂ PL (Γ ⊢ G) ) **

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.

As long as the above sentence accurately sums up the GIT no greater understanding of the GIT is required to totally refute it:

(1) Every WFF of all formal systems must be a truth bearer.

(2) If no formal proof exists within a formal system to show that an expression evaluates to exactly one of the set: {true, false}, then this expression is not a truth bearer, and not a WFF in this formal system.

(3) Therefore there are no WFF in any formal system which can be neither proved nor disproved within this formal system.

]]>**The yellow highlighted pages of the provided book show two things:**

(1) Solving the Liar Paradox is crucially important to fixing the currently inconsistent understanding of the concept of Truth.

(2) Prior to the solution provided on this website it was only understood that the Liar Paradox is defective the precise nature of this defect was unknown.

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