Provable explained in using conventional notation:

Logical_consequence#Syntactic_consequence

**Direct quote from above Wikipedia link:**

A formula ** A** is a syntactic consequence within some formal system

**of a set**

*FS***Γ**of formulas if there is a formal proof in

**of**

*FS***from the set**

*A***Γ**.

**Provable**

_{FS}(Γ, A) // Paraphrase of Γ ⊢_{FS}A**In other words:** Formula A is Provable within Formal System FS if there exists an inference chain (connected sequence of WFF) from a set of Formula Γ to Formula A.

Furthermore if Γ are axioms of FS, then Provable(Γ, A) ≡ True(A) in FS:

**True _{FS}(A) ≡ Provable_{FS} (Γ, A) **

**False**

_{FS}(A) ≡ Provable_{FS}(Γ, ~A)Last paragraph copyright 2017 Pete Olcott