The forcing companions of number theories

This paper is concerned with the finite forcing companion T ^f and the infinite forcing companion T ^F of a number theory T. A number theory is any theory containing the forall _2 - {text{part}} of peano number theory P. Two of our results are as follows: (A) for each number theory T, the theory T ^f is not arithmetical, and the theory T ^F is not analytical, and (B) there is a sentence sigma in forall _4 such that, for each two (not necessarily distinct) number theories T₁, T₂, both σ∈T ₁ ^f and ⌍ σ∈T ₂ ^F .