Abstract

The provability of the axiom of double induction (ADI) with the open induction formula in the additive arithmetic is investigated. The system of additional axioms and theirs provability by ADI is presented.

Highlights

  • In the free variable systems of arithmetic the problem of the replaceability of the axiom of the double induction (ADI) is more interesting in the systems without the restricted difference, as in such systems ADI is definitely stronger than the usual axiom of induction

  • In this paper we will be finish doing to investigate of the question how strong ADI is

  • ∀xA(x, 0)&∀yA(0, y)&∀xy[A(x, y) ⊃ A(x, y )] ⊃ ∀xyA(x, y) can be proved in calculus Z∗ for all open formulae A(x, y) that correspond to the following restriction (κ): the formulae of form mx + q = my + t can enter in a disjunctive normal form of the formula A(x, y) only in such cases: t = q or t, q ∈ N

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Summary

Introduction

In the free variable systems of arithmetic the problem of the replaceability of the axiom of the double induction (ADI) is more interesting in the systems without the restricted difference, as in such systems ADI is definitely stronger than the usual axiom of induction. In this paper we will be finish doing to investigate of the question (which had begun to study in [1] and [2]) how strong ADI is. In the papers [1] and [2] we was shown, that the axiom of the double induction

Results
Conclusion

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