A family of groups with nice word problems

This paper is an outgrowth of my old battle with the open sentence problem for the theory of finite groups. The unsolvability of the word problem for groups (cf. [1] and [4]) entails the undecidability of the open sentence problem for the elementary theory of groups and thus strengthens the original undecidability result for this theory (cf. [7]). The fact that the elementary theory of finite groups is also undecidable (cf. [2] and [6]) therefore justifies my interest in the open sentence problem for that theory. This paper contains a construction of groups that might lead to a negative solution.

Open sentences and the induction axiom

The purpose of this paper is to give a partial answer to the question: How much is the induction axiom weakened if it is applied only to sentences with no bound variables? It is well known that for the full Peano arithmetic this is a weakening ([1] p. 90). We consider Peano arithmetic without multiplication, and give a full answer to the question. It turns out that only four new theorems can be proved from the weakened induction axiom; i.e., all further consequences of this axiom are derivable from these four.We consider a system T formulated within the first-order predicate calculus with equality. The system contains the constant 0 and the three function symbols S (successor), P (predecessor), and +. The non-logical axioms are:A sentence is open if it contains no bound variables. We obtain the system TI from T by adding the rule of inference:(I) If A(x) is an open sentence, infer A(x) from A(0) and A(x) ⊃ A(Sx).The following open sentences are easily proved in TI:The system formed by adding (B1) - (B4) to T is called T′.We abbreviate SS … Sx, where S occurs n times, to Snx. Similarly, we abbreviate (… (x+x)+ …) + x, where x occurs n times, to nx. A term of the form n1x1 + … + nkXk + Sp0 is called simple. An equation between simple terms is called a simple equation.