*L*. Let L be a subset of

_{A}*L*and let T be some AFT of arithmetic.

_{A}We say that T is

**L-sound**iff, for any sentence φ in L, if T ⊢ φ, then φ is true.

We say that T is

**L-complete**iff, for any sentence φ in L, if φ is true, then T ⊢ φ.

In chapter 9, Peter Smith defines a subset of

*L*, called Σ

_{A}_{1}. Then, using the fact that Q is order-adequate, he proves that Q is Σ

_{1}-complete. This is important because the well-formed formulas (

**wff**'s) of Σ

_{1 }can express the decidable numerical properties and relations, and therefore Q will be sufficiently strong. Now to the details...

First, let's define a few interesting subsets of

*L*:

_{A}