On How the Introducing of a New $\theta$ Function Symbol Into   Arithmetic's Formalism Is Germane to Devising Axiom Systems that Can   Appreciate Fragments of Their Own Hilbert Consistency
  A new $\theta$ function primitive is proposed that almost achieves the combined efficiency of the addition, multiplication and successor growth operations. This $\theta$ function symbol enables the constructing of an "IQFS(PA+)" axiom system that can corroborate a fragmentary definition of its own Hilbert consistency, while it will simultaneously verify isomorphic counterparts of all Peano Arithmetic's $\Pi_1$ theorems. Many propositions and intermediate results are also established. Only one intermediate result, which most readers will intuit should be true, does remain formally unproven. 
Dan E. Willard
math.LO