Bounded existential induction

Abstract

The present work may perhaps be seen as a point of convergence of two historically distinct sequences of results. One sequence of results started with the work of Tennenbaum [59] who showed that there could be no nonstandard recursive model of the system PA of first order Peano arithmetic. Shepherdson [65] on the other hand showed that the system of arithmetic with open induction was sufficiently weak to allow the construction of nonstandard recursive models. Between these two results there remained for many years a large gap occasioned by a general lack of interest in weak systems of arithmetic. However Dana Scott observed that the addition alone of a nonstandard model of PA could not be recursive, while more recently McAloon [82] improved these results by showing that even for the weaker system of arithmetic with only bounded induction, neither the addition nor the multiplication of a nonstandard model could be recursive.Another sequence of results starts with the work of Lessan [78], and independently Jensen and Ehrenfeucht [76], who showed that the structures which may be obtained as the reducts to addition of countable nonstandard models of PA are exactly the countable recursively saturated models of Presburger arithmetic. More recently, Cegielski, McAloon and the author [81] showed that the above result holds true if PA is replaced by the much weaker system of bounded induction.However in both the case of the Tennenbaum phenomenon and in that of the recursive saturation of addition the problem remained open as to how strong a system was really necessary to generate the required phenomenon. All that was clear a priori was that open induction was too weak to produce either result.

Tennenbaum's theorem for models of arithmetic

“…A much sharper result using essentially the same ideas was achieved by Wilmers [31] who showed the same result for the subtheory IE 1 of I∆ 0 where induction axioms are only allowed for bounded existential formulas, i.e., formulas of the form ∃ȳ < p (x) q (x,ȳ) = r (x,ȳ) where p, q, r are polynomials with nonnegative integer coeficients. Wilmers achieved some improvements on the above argument, firstly by taking A, B to be disjoint r.e., recursively inseparable sets of primes, but more particularly by using the MRDP theorem on the diophantine representation of r.e.…”

“…(Details are given in Models of Peano Arithmetic [7].) This argument works very well in contexts where overspill is available, and Wilmers [31] shows that SSy (M ) is a Scott set whenever M IE 1 is nonstandard. For weaker theories we still seem to need alternative arguments.…”

Real closures of models of weak arithmetic

“…Does the property "every unbounded rcf with an integer part satisfying T is recursively saturated" hold for T equal to: We note that [Wil85] shows that the reduct of a nonstandard model of IE 1 to the language of + and ≤ is recursively saturated, but the techniques used to prove that result do not seem to be directly applicable in our case. IOpen( x/y ) is a theory about which little is known, except that it is strictly stronger than IOpen ( [Kay93]) and contained in (an extension by definitions of) IE 1 .…”

“…in [HP93]. I∆ 0 and its fragments ( [HP93,Wil85]) are defined as follows (the presentation below is in terms of semirings with a least element rather than rings). Bounded quantifiers are introduced by…”

“…Earlier properties of this kind include the one arising from Tennenbaum's theorem, i.e. "T has no recursive nonstandard models" [Ten59,She64,Wil85,BO96,Moh06], as well as "every nonstandard model of T has a nonstandard initial segment satisfying PA" [McA82,Par84] and "the reduct of every nonstandard model of T to the language of addition is recursively saturated" [CMW82,Wil85]. Of course, each concrete property leads to a different notion of "significant arithmetical content".…”