About the equivalent replaceability of the double induction axiom

In this paper the first order predicate calculus with the axioms of additive arithmetic is investigated. The conditions of the equivalent replaceability of a double induction axiom is presented.


Introduction
In the systems without the restricted difference the axiom of double induction (ADI) is definitely stronger than the usual axiom of induction. J.R. Shoenfield [1] was investigating (by using the models theory) the replaceability of the usual induction rule (RI) with the open induction formula in additive arithmetic, containing the axioms B1-B6 and showed that the axioms B5 and B6 are not provable by RI, but are provable by using the rule of double induction (RDI). J.C. Sheperdson [2] has risen also a question -how strong RDI is.
In this paper we will finish the investigation (which has begun in [3,4] and [5]) of equivalent replaceability of a double induction axiom in the additive arithmetic.

Description of basic calculi
Let Z 0 be the sequential variant of the first order predicate calculus with the signature {= (equality), 0 (zero), (successor), P (predecessor), + (plus)} and usual rules for the logical symbols (see, e.g., [3]); structural rules, cut rule (where , , Z, are finite (probably, empty) sequences of the formulae; the expression α β denotes substitution β for every occurrence of α in every formula of ; F is formula, t is an arbitrary term), axioms A5. → t + s = (t + s) and the axiom of double induction (ADI): LetZ be the system, obtained form Z 0 by replacement of the ADI by the following axioms: We shall call the formulas of the form mx + q = t, my + q = t, mx + ny + q = t, mx + q = ny + t, where x, y are free variables; m = 0, n = 0 are the natural numbers; q, t be a terms, that do not contain the variables x and y, substantial elementary formulas and the formulas of the form x = y ∨ ∃c(x + c = y) ∨ ∃c(x = c + y) -∃-elementary formulas of the calculus Z 0 (also and of the calculusZ).
The formula A(x, y) of the calculusZ we shall call ∃-reduced, if there is of the form (1) where E ij (x, y) are substantial elementary formulas;Ẽ ij (x, y) are ∃elementary formulas or the negations of theirs; D i is a formula that does not contain variables x and y; U, R 1i , R 2i , R 3i (i ∈ U) are finite subsets of the set of natural numbers, and the definition of the set U is detailed in such way: A(x, y) be an ∃-reduced formula of the calculusZ, then 1 Z ADI. A(x, y) is open, then we shall in usual way (see, e.g., [3]) reconstruct it into the disjunctive normal form

Really, if the induction formula
where E ij (x, y) are substantial elementary formulas; U, R 1i , R 2i (i ∈ U) are finite subsets of the set of natural numbers; D i is a formula that does not contain variables x and y.
If A(x, y) is an ∃-reduced formula, its d.n.f. has a shape (1). Let η be the number of the elements of the set U; µ i be the number of the conjuncts in the disjunct B i (x, y) and we shall mark The consequences of the sequent where A(a, b); a, b be parameters that do not occur in ; ν, τ, ε, ω, ξ ∈ N; ω = ε; ν l = ν k , τ l = τ k , if l = k; l, k ∈ N .
For the sequent (2) are possible the following cases: i ∈ U 1 or i ∈ U 2 , then proof of the sequent (2) is constructed analogously, as in [4] (see Lemma 2 and Theorem 2). If i ∈ U 3 , then R 1i = R 2i = ∅ and whereẼ ij (x, y) are an ∃-elementary formulas or the negations of theirs; D i is a formula that does not contain variables x and y. Let 1 The expression Z Q will denote, that the object Q is deducible in the calculus Z.
THEOREM 2. The calculus Z 0 andZ are equivalent: Proof. Part I. The axiom ADI with ∃-reduced induction formula A(x, y) is provable in the calculusZ by Theorem 1.
Part II. We must show that the all axioms of the calculusZ are provable in the calculus Z 0 . The axioms B1-B4 are provable by the usual induction axiom (see, e.g., [1,2]), so they all the more are provable by the ADI. The axioms B5, B6 can not be proved by the usual induction axiom, we can prove them by ADI (see [4]). Let then basis for the axiom B7 is the sequent → A(t, 0), i.e., → t = 0 ⊃ (∃r(t + r = 0) ∨ ∃ω(t = ω )), and from that follow → t = 0 ⊃ t = (P t) , i.e., the axiom B1. The provability of the sequent → A(0, s) is showed analogously.
The all of theirs can be reduced to the sequents → t = t ; →s = s and → t = t accordingly, i.e., to the axiom P2.