J. K. Truss

The group of the countable universal graph

Let C be a set with at least two, and at most ℵ 0 , members, and for any set X let [ X ] 2 denote the set of its 2-element subsets. If Γ is a countable set, and F c is a function from [Γ] 2 into C, then the structure Γ c = (Γ, F c ) is called the countable universal C-coloured graph if the following condition is satisfied: Whenever α is a map from a finite subset of Γ into C there is x eΓ–dom α such that (∀ y edom α) F c { x, y } = α(y).

Sets Having Calibre

Publisher Summary It is well-known that if the complete Boolean algebra satisfies the countable chain condition (c.c.c.) then cardinals are preserved in a Boolean extension(similarly for forcing, using a partially ordered set with greatest element). However the c.c.c. has its disadvantages, the main from the point of view being that the Boolean algebra may satisfy the c.c.c. in V without satisfying it in the Boolean extension. This occurs for example if Boolean algebra is a Souslin algebra. Thus, one leads to consider stronger notions, which have more satisfactory closure and extension properties. The chapter discusses the general properties of sets having caliber. A typical such property is that if Boolean algebra has calibre then it still has caliber in any c.c.c. extension. The chapter is concerned with certain specific Boolean algebras, notably the “amoeba” and “dominating” algebras, which are good examples of algebras having caliber, and which have considerable importance for the study of Lebesgue measure and Baire category on the real line.

First Steps into Metapredicativity in Explicit Mathematics

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Generic automorphisms of the universal partial order

We show that the countable universal-homogeneous partial order (P,<) has a generic automorphism as defined by the second author, namely that it lies in a comeagre conjugacy class of Aut(P,<). For this purpose, we work with ‘determined’ partial finite automorphisms that need not be automorphisms of finite substructures (as in the proofs of similar results for other countable homogeneous structures) but are nevertheless sufficient to characterize the isomorphism type of the union of their orbits.

On ℵ0-categorical weakly o-minimal structures☆

Abstract ℵ 0 -categorical o -minimal structures were completely described by Pillay and Steinhorn (Trans. Amer. Math. Soc. 295 (1986) 565–592), and are essentially built up from copies of the rationals as an ordered set by ‘cutting and copying’. Here we investigate the possible structures which an ℵ 0 -categorical weakly o -minimal set may carry, and find that there are some rather more interesting (and not o -minimal) examples. We show that even here the possibilities are limited. We subdivide our study into the following principal cases: the structure is 1-indiscernible, in which case all possibilities are classified up to binary structure; the structure is 2-indiscernible, classified up to ternary structure; the structure is 3-indiscernible, in which case we show that it is k -indiscernible for every finite k . We also make some remarks about the possible structures of higher arities which an ℵ 0 -categorical weakly o -minimal structure may carry.

The structure of amorphous sets

Abstract A set is said to be amorphous if it is infinite, but is not the disjoint union of two infinite subsets. Thus amorphous sets can exist only if the axiom of choice is false. We give a general study of the structure which an amorphous set can carry, with the object of eventually obtaining a complete classification. The principal types of amorphous set we distinguish are the following: amorphous sets not of projective type, either bounded or unbounded (depending on whether there is a bound on the (predominant) size of members of partitions of the set into finite pieces), and amorphous sets of projective type, meaning that the set admits a non-degenerate pregeometry, over finite fields either of bounded cardinality or of unbounded cardinality. The hope is that all amorphous sets will be of one of these types. Examples of each sort are constructed, and a reconstruction result for bounded amorphous sets is presented, indicating that (under certain set-theoretic assumptions) the amorphous sets of this kind constructed in the paper are the only possible ones. The final section examines some questions concerned with the resulting cardinal arithmetic.

Countable homogeneous multipartite graphs

We give a classification of all the countable homogeneous multipartite graphs. This generalizes the similar result for bipartite graphs given in Goldstern et al. (1996) [6].

On computable automorphisms of the rational numbers

The relationship between ideals I of Turing degrees and groups of I -recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I , and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.

On k-CS-Transitive Cycle-Free Partial Orders with Finite Alternating Chains

The notion of cycle-free partial order (CFPO) was defined by R. Warren, and the major cases of the classification of the countable sufficiently transitive CFPOs were given, the finite and infinite chain cases, by Creed, Truss, and Warren. It is the purpose of this paper to complete the classification. The cases which remained untreated were CFPOs not embedding an infinite ‘alternating chain’ ALT (which can only happen in the finite chain case). It is shown that if a k-CS-transitive CFPO does not embed ALT, where k ≥ 3, then it does not embed any alternating chain of size k + 3, and this leads to the desired classification (which is only given explicitly for k = 3 and 4). The general result says that the class of k-CS-transitive CFPOs for k ≥ 3 not embedding ALT admits a recursive classification.

On Homogeneous Semilattices and Their Automorphism Groups

We show that there are just countably many countable homogeneous semilattices and give an explicit description of them. For the countable universal homogeneous semilattice we show that its automorphism group has a largest proper nontrivial normal subgroup.