The next meeting is on tuesday 8th march. Venue: Centre for Mathematical Sciences, Wilberforce Road, Cambridge. We have booked Meeting Room 10 from 0900 to 1300 and Meeting Room 4 from 1300 to 1730. After that there will be drinks and nibbles in a venue yet to be determined.

We have four speakers. Aspero is the first speaker by special request, at 1200 in MR 10, the Mary Godfrey Room on the first floor of Pavilion B; the other talks are not yet timetabled, but my guess is that subsequent talks will be at 1315, 1430, 1545 and 1700, tho' this is not yet set in stone

Forcing simply definable well-orders of the reals

ZFC, the usual system of axioms of set theory, with the Axiom of Choice, certainly implies that there is a well-order of the reals. However it is not difficult to see that it does not imply the existence of such a definable well-order. A consequence of results of Woodin is that, in the presense of sufficiently strong large cardinals, there cannot be any well-order of the reals that admits a ZFC-provably antisymmetric definition over $H (\omega_2)$ of complexity $\Sigma_2$ or $\Pi_2$ (without parameters). In this talk I will look at the possibility of forcing a well-order of the reals that admits ZFC-provably antisymmetric $\Sigma_3$ and $\Pi_3 $ definitions over $H(\omega_2)$ without parameters. In view of the above mentioned limitations this is then the optimal result from the point of view of the Levy hierarchy of formulas (if sufficiently strong large cardinals exist). Our starting hypothesis will be the existence of an inaccessible limit of measurable cardinals and the construction will use a technique for manipulating the guessing or non- guessing of all canonical functions for ordinals in $\omega_2$ by a single function $H$ defined on a stationary subset of $\omega_1$ such that, for all $\nu\in dom(H)$, $H(\nu)$ is a subset of $\omega_1$ of suitably bounded order type. The existence or nonexistence of such functions $H$ defines a possibly new family of combinatorial properties for subsets of $\omega_1$.

Cuts and gluts THE ABSTRACT

We define validity with respect to models allowing gluts and connect this with the proof-theoretical phenomenon of cut-absorption, by defining a notion of glut-derivation and showing that a sentence is valid in the new sense iff it is glut-derivable. This part includes a systematisation of well known results by Schutte and Girard. We end by providing an example of a set-theoretic system enjoying cut elimination without extensionality, but not when an extensionality rule is present.

Open colourings and cardinal invariants.

Abstract:

We show that the Open Colouring Axiom introduced by Abraham, Rubin and Shelah (OCA_[ARS]) implies that most of the classical cardinal invariants of the continuum, i.e., the ones in Cichon's diagram, are big. We will also mention a dual of OCA_[ARS].

Abstract

The subject of this talk is positive abstraction in set theory. By "abstraction" (instead of comprehension), it is understood that the language is equipped with an abstractor "{-|-}" which allows the formation of set abstracts as primitive terms; and by "positive" it is meant that the use of the abstractor is restricted to negation-free formulas in that extended language. As for the untyped lambda calculus, the consistency of positive abstraction with extensionality can be proved by a term model construction or by means of Scott-style models. We shall focus on these latter here to present some unexpected consequences of the use of an abstractor in (positive) set theory.

Cameleon: May 2005 in Cambridge