Line 1 I to the power of ts sub kn sub o to the power of wn to the power of tht surd Line 2 N e
minus beta Line 3 l og N ll r open parenthesis N closing parenthesis ll N open parenthesis lo …

… model has sets of integers, including negative numbers, as … every hyper-arithmetical set of
numbers. The base of the … every recursive set by equations over sets of natural numbers with …

… The base of the construction is the authors’ earlier result [9] on representing every recursive
set by equations over sets of natural numbers with union and addition. In Sect. 2, this result …

… This book is dedicated to a very interesting branch of the theory of numbers, the solution
of equations in integers. The determination of integral solutions of algebraic equations with …

… A set of positive integers %, u2, ... will be called an ji-set if no three of the … For if ai^-j-6,
au2-\-b, ... form an j4-set then so do ux, u2, ..., and conversely. Further, for any two positive integers …

… Relatively many sum-free sets can be obtained by allowing a few numbers just smaller
than n. Our main result is that this does not produce enough sets to violate the conjecture. …

… If E is a set of (conditional) equations over 2 and 'l is a 2-algebra such that T) = E, we say
that U is an E-algebra. We define ALG(2, E) as the class of all E-algebras and T(2, E) as the …

… set a+q2u,, a+q2u~ ..... a+q2u~ (where a is an integer and t, q, u~, us, ..., ut are positive
integers) is an d-set relative to the set of the squares if and only if also the set u,, u2, ..., u, has this …

… The goal of this section is to prove one half of the main theorem: if a set of integers is E„
definable then it is El over the least nreflecting admissible ordinal. The proof uses the normal …

… equations over sets of natural numbers by Proposition 1. This construction is replicated for
equations over sets of integers… The constant sets of negative integers in Lemmata 1 and 2 are …