2/November/2012

**11:00–12:00, Fields,Room 230**

Speaker: A.R.D. Mathias

Title: The truth predicate and the forcing theorem in weak subsystems of ZF

Abstract: Devlin in his book “Constructibility” sought a theory true in every limit Goedel fragment $L_{\omega\nu}$ and every Jensen fragment $J_\nu$ (where $\nu\ge 1$) and strong enough to define the truth predicate for $\Delta_0$ formulae.

For some years I sought to identify the weakest fragment of ZF that would support a recognisable theory of set forcing, and in particular the definition of $p\Vdash \phi$ for $\phi$ a $\Delta_0$ formula.

These two quests turn out to have common ground and have resulted in the theory of rudimentary recursion and provident sets, which will be described in the talk.

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**13:30–15:00, Fields,Room 230**

Speaker: Antonio Avilés

Title: $P(\omega)/fin$ and its close relatives

Abstract: We shall discuss uncountable Fraisse limits and iterated push-out constructions. This is related to the problem of finding structural characterizations of the Boolean algebra $P(\omega)/fin$ like Parovichenko’s theorem under CH, and Dow–Hart‘s characterization in the model obtained by adding $\aleph_2$ Cohen reals to a model of CH. We shall explore some connections with Banach spaces as well.