Finite Reverse Mathematics

We present some formal systems in the language of linearly ordered rings with finite sets whose nonlogical axioms are strictly mathematical, which correspond to polynomially bounded arithmetic. With an additional strictly mathematical axiom, the systems correspond to exponentially bounded arithmetic. 1. T0 and IS0. In this section, we introduce the system T0, and show that it corresponds to the system IS0 of polynomially bounded arithmetic (presented below). Let T0 be the following system in the two sorted language with variables over integers and variables over finite sets of integers. For the integer sort, we use the language 0,1,+,,•,<,= of linearly ordered rings. We use Œ between integers and sets. Equality is used only between integers. The official integer variables are x0,x1,..., and the official set variables are A0,A1,... . The nonlogical axioms of T0 are as follows. 1. Linearly ordered ring axioms. 2. Finite interval. ($A)("x)(x Œ A ́ (y < x Ÿ x < z)). 3. Boolean difference. ($C)("x)(x Œ C ́ (x Œ A Ÿ ÿ(x Œ B))). 4. Set addition. ($C)("x)(x Œ C ́ ($y)($z)(y Œ A Ÿ z Œ B Ÿ x = y+z)). 5. Set multiplication. ($C)("x)(x Œ C ́ ($y)($z)(y Œ A Ÿ z Œ B Ÿ x = y•z)). 6. Least element. ($x)(x Œ A) Æ ($x)(x Œ A Ÿ ÿ($y)(y Œ A Ÿ y < x)). The linearly ordered ring axioms are as follows.