The next meeting is on thursday 26 May. Venue: Meeting Room 11, Centre for Mathematical Sciences, Wilberforce Road, Cambridge.
Speakers (so far). Thomas Forster, Aldo Antonelli, John Truss, Imre Leader. Times are extremely flexible.
Title: The Homomorphism Problem for $\beta N$.
Abstract: The space $\beta N$ of ultrafilters on the natural numbers has a natural topological and algebraic structure. One of the most fascinating of the many unsolved problems about $\beta N$ is the `homomorphism problem': does there exist a non-trivial homomorphism from $\beta N$ to itself? It turns out that this reduces to a very down-to-earth question about ultrafilters. After giving some background, we will discuss recent progress, including some links with a curious combinatorial problem.
Title: FIRST-ORDER NUMERICAL ABSTRACTION
There are two main reduction strategies for arithmetic: according to the first strategy, originating with Frege and Russell, the natural number are identified with equivalence classes under the equinumerosity relation. The second strategy, championed by Zermelo and von Neuman, rather identifies particular representatives of those equivalence classes. The former approach is usually thought to be more general, but suffers the drawback of being intrinsically higher-order; the latter approach can be carried out at the first-order but at the price of identifying the natural numbers with a particular kind of entities.
Here we present a way to carry out the Frege-Russell reduction of arithmetic at the first-order. Our main tool is the introduction of a slight (?) generalization of the standard first-order quantifiers, "there is" and "for all". In particular, after introducing the first-order (!) binary quantifier Fx(A,B) stating that there are no more As than BS, along with a heterogenous predicate cosnstant N(x,A) to the effect that x numbers the As, we propose arithmetical axioms for such a language, exploring exactly how much of arithmetic can be captured in the resulting system.
Generic Automorphisms and Circular orders
J. K. Truss
Generic automorphisms of three particular homogeneous structures are considered, as an ordered set, the countable universal homogeneous partial order, and the random graph. I study the existence of mutually generic automorphisms, and three possible definitions of what this could mean.
The next meeting will be in Cambridge on thursday 23rd or fri 24th November. Speakers will include Greg Restall of ANU and Allen Mann of Colorado.
November 2005 in Cambridge