Hyperuniverses
by
Furio Honsell
Universitá di Udine
Coauthors: M. Forti, M. Lenisa

Hyperuniverses ([2]) are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in Computer Science ([3]), for giving denotational semantics à la Scott-de Bakker, and in Mathematical Logic ([1]), in order to show the consistency of set theories which do not abide by the ``limitation of size'' principle.

Definition 1 Let U be a set of atoms. A set N subset or equal P(N) \cup U is a \kappa-hyperuniverse if the complements, relative to N, of the elements of N \cap P(N) are a \kappa-additive, \kappa-compact topology \tau on N, such that N is the disjoint union of U and the space of its closed subsets endowed with Vietoris \kappa-topology.

We give existence theorems for hyperuniverses and discuss applications and generalizations to the non \kappa-compact case. In particular we prove the following axiomatic characterization:

Theorem 1 A set N subset or equal P(N) \cup U, such that \kappa = min{ |X| | X subset or equal N , X\not in N }, is a \kappa-hyperuniverse if and only if: \emptyset, N in N; if X, Y in N then {X, Y} in N; if X in P(N) then \cap X in N; if X in N \cap P(N) then X \cap P(N), {Y in N | X \cap Y =/= \emptyset}, \cup X in N; if x, Y in N and x not in Y then existsZ, X in N. such that Z \cup X = N, x\not in Z and Y \cap X = \emptyset.

[1]

M.Forti, R.Hinnion: ``The consistency problem for positive comprehension principles'', J.S.L. 54 (1989).

[2]

M.Forti, F.Honsell: ``A General Construction of Hyperuniverses'', Theor. Comp. Sci. 156 (1996).

[3]

M.Forti, F.Honsell, M.Lenisa: ``Processes and Hyperuniverses'', Springer L. N. Comp. Sci. 841 (1994).

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-39.