The results in this paper were motivated by the following result due to R. Solovay.

Theorem 1 (Solovay). Let M be a nonstandard model of Peano's first order axioms P and let I ⊂e M (i.e. ϕ ≠ ⊂ M and I is closed under < and successor). Then for each of the functions we can define J ⊆e I in ‹M, I› such that J is closed under that function. (∣x∣ denotes [log2(x)].)

Proof. Just notice that the cuts defined by

are successively closed under

In view of Theorem 1, the following question was raised by R. Solovay: Can we define J ⊆ I in ‹M, I› such that J is closed under exponentiation? In Theorem 2 we show that the answer is “no”. Theorem 3 is based on Theorem 2 and extends the technique to cuts which are models of subsystems of P.

To prove both theorems we shall need an estimate due to R. Parikh (see [1], especially the proof of Theorem 2.2a). For the sake of completeness, and also to introduce some notation we shall sketch Parikh's estimate in the next section. At all times we shall give the easiest estimates which still work rather than the sharpest ones.

There exists a model in a countable language having a unique element which is not definable in .

This paper describes Tarski’s project of rehabilitating the notion of truth, previously considered dubious by many philosophers. The project was realized by providing a formal truth definition, which does not employ any problematic concept.

We consider the monadic second-order theory of linear order. For the sake of brevity, linearly ordered sets will be called chains.

Let = ⟨A <⟩ be a chain. A formula ø(t) with one free individual variable t defines a point-set on A which contains the points of A that satisfy ø(t). As usually we identify a subset of A with its characteristic predicate and we will say that such a formula defines a predicate on A.

A formula (X) one free monadic predicate variable defines the set of predicates (or family of point-sets) on A that satisfy (X). This family is said to be definable by (X) in A. Suppose that is a subchain of = ⟨B, <⟩. With a formula (X, A) we associate the following family of point-sets (or set of predicates) {P : P ⊆ A and (P, A) holds in } on A. This family is said to be definable by in with at the background.

Note that in such a definition bound individual (respectively predicate) variables of range over B (respectively over subsets of B). Hence, it is reasonable to expect that the presence of a background chain allows one to define point sets (or families of point-sets) on A which are not definable inside .

Let ψ be a partial recursive function (of one argument) with λ-defining term F∈Λ°. This means There are several proposals for what F⌜n⌝ should be in case ψ(n) is undefined: (1) a term without a normal form (Church); (2) an unsolvable term (Barendregt); (3) an easy term (Visser); (4) a term of order 0 (Statman).

These four possibilities will be covered by one ‘master’ result of Statman which is based on the ‘Anti Diagonal Normalization Theorem’ of Visser (1980). That ingenious theorem about precomplete numerations of Ershov is a powerful tool with applications in recursion theory, metamathematics of arithmetic and lambda calculus.

We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.

For any subset $Z \subseteq {\mathbb {Q}}$, consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$, we show that if Z is not thin in ${\mathbb {Q}}$, then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $-definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

We prove that the class of trees with no branches of cardinality ≤ κ is not RPC definable in L∞κ when κ is regular. Earlier such a result was known for under the assumption κ<κ = κ. Our main result is actually proved in a stronger form which covers also L∞κ (and makes sense there) for every strong limit cardinal λ < κ of cofinality κ.

In this article we explain two different operational interpretations of functional programs by two different logics. The programs are simply typed λ-terms with pairs, projections, if-then-else and least fixed point recursion. A logic for call-by-value evaluation and a logic for call-by-name evaluation are obtained as as extensions of a system which we call the basic logic of partial terms (BPT). This logic is suitable to prove properties of programs that are valid under both strict and non-strict evaluation. We use methods from denotational semantics to show that the two extensions of BPT are adequate for call-by-value and call-by-name evaluation. Neither the programs nor the logics contain the constant ‘undefined’.

This article examines the categorical problem that persons of ‘mixed-race’ background presented to British administrations in eastern, central and southern Africa during the late 1920s and 1930s. Tracing a discussion regarding the terms ‘native’ and ‘non-native’ from an obscure court case in Nyasaland (contemporary Malawi) in 1929, to the Colonial Office in London, to colonial governments in eastern, central and southern Africa, this article demonstrates a lack of consensus on how the term ‘native’ was to be defined, despite its ubiquitous use. This complication arrived at a particularly crucial period when indirect rule was being implemented throughout the continent. Debate centered largely around the issue of racial descent versus culture as the determining factor. The ultimate failure of British officials to arrive at a clear definition of the term ‘native’, one of the most fundamental terms in the colonial lexicon, is consequently suggestive of both the potential weaknesses of colonial state formation and the abstraction of colonial policy vis-à-vis local empirical conditions. Furthermore, this case study compels a rethinking of contemporary categories of analysis and their historical origins.