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# Minimality and completions of PA

## Extract

The results in this paper say that natural upper bounds for sets of degrees associated with theories and models of arithmetic cannot be minimal. The basic new result says that for any completion T of PA, there is another completion S such that S<TT and Rep(S) = Rep(T). This immediately implies that deg(T) is not minimal over {deg(X): X ∈ Rep(T)}. As an application of the basic result, we obtain the fact that if is a non-standard model of TA (true arithmetic), then deg () cannot be minimal over {deg(X): X is arithmetical}. More generally, if is a non-standard model of an arbitrary completion T of PA, then deg() cannot be minimal over {deg(): X ∈ Rep(T)}. We vary the basic result, making S′ ≡TT′. As an application of the variant, we obtain the fact that if is a non-standard model of PA, then {deg(): } has no minimal element.

The remainder of the present section gives a brief account of the background needed for the basic new result and the variant. These two results are proved in Section 2. The applications are given in Section 3, along with further background needed for the applications. One important source of ideas used in the present paper is a paper of Scott [9]. In addition, there are ideas taken from Tennenbaum [12], Feferman [3], Marker [6], [7], and Solovay. Chapter 19 of [1] gathers together most of this material. In fact, it contains all that is really essential. In one application, we appeal to Solovay's result on degrees of models of an arbitrary completion of PA, a result which is not completely proved in [1]. However, for the best application, which implies all the others, we use only some ideas from the proof of Solovay's theorem. These are given in lemmas that are proved in Section 3, where they are needed, or taken from [1], While the proof of Solovay's result requires an infinitely nested priority construction, our best application rests on nothing more than finite-injury constructions.

## References

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[1]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, 2000.
[2]Davis, M., Hilbert's tenth problem is unsohable, American Mathematics Monthly, vol. 80 (1973), pp. 233269.
[3]Feferman, S., Arithmetical definable models of formalizable arithmetic, Notices of the American Mathematical Society, vol. 5 (1958), p. 679.
[4]Knight, J. F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), pp. 10341042.
[5]Knight, J. F., True approximations and models of arithmetic, Models and computability (Cooper, B. and Truss, J., editors), Cambridge University Press, 1999, pp. 255278.
[6]Macintyre, A. and Marker, D., Degrees of recursively saturated models, Transactions of the American Mathematical Society, vol. 282 (1984), pp. 539554.
[7]Marker, D., Degrees of models of true arithmetic, Proceedings of the herbrand symposium (Stern, J., editor), North-Holland, 1981, pp. 233242.
[8]Matijasevic, Yu., On recursive unsolvability of hilbert's tenth problem, Proceedings of the fourth international congress on logic, methodology, and philosophy of science, bucharest, 1971, North-Holland, 1973, pp. 89110.
[9]Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Dekker, , editor), American Mathematical Society, 1962, pp. 117–22.
[10]Solovay, R., unpublished manuscript, circulated in 1982.
[11]Solovay, R., personal correspondence, 1991.
[12]Tennenbaum, S., Non-archimedean models for arithmetic, Notices of the American Mathematical Society, (1959), p. 270.

# Minimality and completions of PA

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