Groupe de Contact
20 septembre 2002 Campus de la Plaine,ULB

Thème: Usual Sets and Models

 

PROGRAMME - ABSTRACTS

MATIN : local 2N0-906
 09h30 T.Forster (Cambridge) : Stratified fragments of ZF
11h00 CAFE
11h30 M.Forti (Pise) : The standard Universe and its Nonstandard Enlargements
 
13h15 repas offert aux participants
ATTENTION : il est impératif de s'inscrire à l'adresse rhinnion@ulb.ac.be, AVANT le 15 septembre.
 
APRES-MIDI : local 2NO-707
 
14h30 R.Holmes (Boise, US) : The double extension set theory of Kisielewicz is inconsistent
16h00 CAFE
16h30 A.Kisielewicz (Wroclaw, Pologne) : Double extension sets as foundation for mathematics

Si vous désirez participer à cette réunion, et prendre part au repas, faites-le savoir (si possible par email) à

Roland Hinnion

rhinnion@ulb.ac.be

VOTRE INSCRIPTION AU LUNCH DOIT NOUS PARVENIR AU PLUS TARD LE 15 SEPTEMBRE .

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O Je participerai au lunch

ABSTRACTS

Thomas Forster (Cambridge) : Stratified fragments of ZF
Two constructions are given of structures which naturally model the stratified fragment of ZF. One (HS) is an analogue of the cumulative hierarchy of Von Neumann, and the other (S) is an analogue of L. AC fails in both. I am indebted to Randall Holmes, Adrian Mathias and Philip Welch for helpful discussions.

Marco Forti (Pise) : TBA

Randall Holmes (Boise,US) : The double extension set theory of Kisielewicz is inconsistent
Andrzej Kisielewicz has presented in two papers versions of a "double extension set theory" proposed as a solution to the paradoxes, and has shown that this system interprets ZFC. In this talk, it will be shown that this system (in any of the versions proposed) is inconsistent. Kisielewicz's system attempts to avoid paradox by providing two different membership relations, with comprehension axioms providing that formulas in terms of either membership relation define extensions for the other membership relation. This appears to avert the Russell paradox. The inconsistency is obtained by showing that any of the systems of double extension set theory proposed can simulate the construction of a fixed point combinator for an operator on finite vectors of truth values with period greater than 2 (the double extension approach prevents the construction of a fixed point combinator for a period 2 operation like negation). Technical difficulties are present in that it is not easy to show that there is a suitable ordered pair to support the interpretation of functions in the set theory, but these difficulties are overcome. If time permits, a related system which is consistent but quite weak will be discussed.

Andrzej Kisielewicz (Wroclaw,Pologne) : Double extension sets as foundation for mathematics
I will discuss the general idea of sets with double extension as a foundation for mathematics and present a new proposal how to avoid the inconsistency discovered recently by Holmes.

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