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# Tennenbaum's theorem and recursive reducts

## Summary

To honor and celebrate the memory of Stanley Tennenbaum

Stanley Tennenbaum's influential 1959 theorem asserts that there are no recursive nonstandard models of Peano Arithmetic (PA). This theorem first appeared in his abstract [42]; he never published a complete proof. Tennenbaum's Theorem has been a source of inspiration for much additional work on nonrecursive models. Most of this effort has gone into generalizing and strengthening this theorem by trying to find the extent to which PA can be weakened to a subtheory and still have no recursive nonstandard models. Kaye's contribution [12] to this volume has more to say about this direction.

This paper is concerned with another line of investigation motivated by two refinements of Tennenbaum's theorem in which not just the model is nonrecursive, but its additive and multiplicative reducts are each nonrecursive. For the following stronger form of Tennenbaum's Theorem credit should also be given to Kreisel [5] for the additive reduct and to McAloon [26] for the multiplicative reduct.

Tennenbaum's Theorem. If M = (M, +, ·, 0, 1, ≤) is a nonstandard model of PA, then neither its additive reduct (M, +) nor its multiplicative reduct (M, ·) is recursive.

What happens with other reducts? The behavior of the order reduct, as is well known, is quite different from that of the additive and multiplicative reducts. The order type of every countable nonstandard model is ω + (ω* + ω) · η, where ω and η are the order types of the nonnegative integers ℕ and the rationals ℚ, respectively.

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