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A recursive model for arithmetic with weak induction

Published online by Cambridge University Press:  12 March 2014

Zofia Adamowicz
Affiliation:
Instytut Matematyczny Pan, Warsaw, Poland
Guillermo Morales-Luna
Affiliation:
Instytut Matematyczny Pan, Warsaw, Poland

Extract

Before stating the results we would like to thank the referee for reorganizing the whole paper and changing its original logical terminology to an algebraical terminology.

Let Z* be a proper elementary extension of the integral domain Z. For any bZ*∖Z we define Zb〉 as the following subring of Z*:

It is easy to prove that the natural divisibility relation on Zb〉 is the restriction to Zb〉 of the divisibility on Z* (see Lemma 1.1 below). The much stronger statement that Zb〉 is algebraically closed in Z* does not hold for all b. Similarly, there are many b's such that Zb〉 is not a recursive ring, e.g. if the set of standard primes dividing b is not recursive then Zb〉 cannot be a recursive ring under any bijection with N.

Our main result is the following.

Theorem. There exists an element bZ*∖Z such that

(1) Zbis algebraically closed in Z*,

(2) Zbis a recursive ring, and

(3) each infinite element of Zbis divisible by a standard integer > 1 (so the only “primes” of Zbare the standard ones).

An easy consequence of (1) is that certain induction axioms are satisfied by Zb〉. This explains the title of the paper. In the last section of the paper, §3, we shall say more about this.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

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