Groupe de Contact 18 et 19 septembre 2003 Campus de la Plaine, ULB Thème: Non-classical sets

PROGRAMME - ABSTRACTS

IMPORTANT: Afin de procéder en temps utile aux réservations
indispensables, les participants sont priés de signaler, à l'adresse
de contact rhinnion@ulb.ac.be et avant le 5 septembre:

1. S'ils participent au lunch de midi du 18 septembre (gratuit pour les personnes inscrites)

2. S'ils participent au repas du soir du 18 septembre (payement sur place)

3. S'ils participent au repas du soir du 19 septembre;

si oui, prière de virer 27 euros sur le compte 068-2085537-64, avec la communication
"Repas UAE".

Date limite d'inscription / virement le 5 septembre 2003. Merci de
votre compréhension.

ABSTRACTS

R. Hinnion:Partial sets.
Paul Gilmore's pioneer work about partial sets is at the origin of several related lines of research,in particular positive set theories and paraconsistent set theories. While for these one has many satisfying results ,the partial sets themselves present much more difficultie; for example is it still not known whether extensionality is compatible with the natural first-order versions.Now is it so that perhaps extensionality is not really the adequate identification criterion in term models for partial set theories, and Gilmore himself developed arguments in favour of intensionality (instead of extensionality).It seems that some "forcing" techniques could bring more information about that;this will be illustrated for (classical) positive sets,with the hope that adaptations to the partial case can be found in the future...

T. Forster:
The assertion $(\forall n \in N)(n \leq Tn)$ is equivalent to the assertion that there is a permutation model containing a partition of the hereditarily finite sets according to set-theoretic rank. This result can be generalised to show that, for any ordinal $\alpha$, the following are equivalent:
 (i) $\beta \leq T\beta$ holds below $\alpha$
(ii) there is a permutation model containing the partition according to (set-theoretic) rank of the collection of sets hereditarily of power less than $\kappa$, as long as the sup of those ranks is less than or equal to $\alpha$

Olivier Esser:
 A model of paraconsistent set theory. The basic idea is to recover the whole comprehension scheme of Frege: For any formula Phi, there exists the set {x | Phi(x)} inside of paraconsistent logic (where we have three truth values: true, false, inconsistent). We will present here a model which contains the ordinary set theory ZF and much more. This is closely related of positive set theory but we can also see connections with "double extension set theory" of A. Kisielewicz (cf. his talk). It is "apparently" dual to the partial set theory problems but the duality works not so good as one could expect (cf. the talk of R. Hinnion).

A. Kieselewicz:
 Double extension set theories. I will discuss the consequences of recent results by Holmes, who proved that stronger systems of double extension set theory are incosnsistent, while in the weakest system different sets with the same extension have to exist. I will show that stronger systems may be still considered after slight modifications forced by Holmes result. I will argue also that Holmes sketch of the proof that ZF can be interpreted in double extension set theories does not seem to work.

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