Timothy J. Carlson

A Dual Form of Ramsey's Theorem

Abstract Let k ϵ ω , where ϵ is the set of all natural numbers. Ramseys Theorem deals with colorings of the k -element subsets of ω. Our dual form deals with colorings of the k -element partitions of ω. Let ( ω ) k (respectively ( ω ) ω ) be the set of all partitions of ω having exactly k (respectively infinitely many) blocks. Given X ϵ ( ω ) ω let ( X ) k be the set of all Y ϵ ( ω ) k such that Y is coarser than X. Dual Ramsey Theorem . If ( ω ) k = C 0 ∪ … ∪ C t −1 where each C i is Borel then there exists X ϵ ( ω ) ω such that ( X ) k ⊆ C i for some i l . Dual Galvin-Prikry Theorem . Same as before with k replaced by ω. We also obtain dual forms of theorems of Ellentuck and Mathias. Our results also provide an infinitary generalization of the Graham-Rothschild “parameter set” theorem [ Trans. Amer. Math. Soc. 159 (1971), 257–292] and a new proof of the Halpern-Lauchli Theorem [ Trans. Amer. Math. Soc. 124 (1966), 360–367].

An infinitary version of the Graham-Leeb-Rothschild theorem

Abstract This paper contains a proof of a conjecture of S. Simpson which is an infinitary version of a conjecture of Rota concerning partitions of finite dimensional vector spaces over a finite field. Rotas conjecture was proved by Graham, Leeb, and Rothschild (Adv. in Math. 8 (1972), 417–433).

Elementary patterns of resemblance

We will study patterns which occur when considering how Σ1-elementary substructures arise within hierarchies of structures. The order in which such patterns evolve will be seen to be independent of the hierarchy of structures provided the hierarchy satisfies some mild conditions. These patterns form the lowest level of what we call patterns of resemblance. They were originally used by the author to verify a conjecture of W. Reinhardt concerning epistemic theories (see Carlson, Arch. Math. Logic 38 (1999) 449–460; Ann. Pure Appl. Logic, to appear), but their relationship to axioms of infinity and usefulness for ordinal analysis were manifest from the beginning. This paper is the first part of a series which provides an introduction to an extensive program including the ordinal analysis of set theories. Future papers will conclude the introduction and establish, among other things, that notations we will derive from the patterns considered here represent the proof-theoretic ordinal of the theory KPl0 or, equivalently, Π11−CA0 (as KPl0 is a conservative extension of Π11−CA0).

A basic unit of computation in distributed systems

The authors define basic units of computation in distributed systems, whether communicating synchronously or asynchronously, as comprising indivisible logical units of computation that take the system from one ground state to another. It is explained how a computation can be viewed as a partial order over the basic units of the computation. The problem of detecting the basic units is considered. One algorithm for creating ground states during a computation in an asynchronously communicating system with FIFO channels is given, and an existing algorithm that implicitly creates ground states in a synchronously communicating system is referenced. The significance of the basic unit is explained, and its applications are given.<<ETX>>

Topological Ramsey Theory

We survey the interplay between topology and Ramsey Theory which began with Ellentuck’s Theorem (Ellentuck 1974) (and was anticipated by work of Nash-Williams (1965), Galvin and Prikry (1973) and Silver (1970) by giving a fairly abstract treatment of what have become known as Ellentuck type theorems.

Knowledge, machines, and the consistency of Reinhardt's strong mechanistic thesis☆

Abstract Reinhardts strong mechanistic thesis, a formalization of “I know I am a Turing machine”, is shown to be consistent with Epistemic Arithmetic.

Ordinal arithmetic and \(\Sigma_{1}\)-elementarity

Abstract. We will introduce a partial ordering

Patterns of resemblance of order 2

\preceq_1

Generalizing Kruskal's theorem to pairs of cohabitating trees

on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use

Tracking chains of Σ2-elementarity☆

\preceq_1