*Wissenschaftliches Programm* † *Liste der Vortragenden*

Sektion 16
Donnerstag, 21.09.2000, 15.00–15.20 Uhr, WIL C 307

An Axiom Capturing the Combinatorics in Cohen Models

Stefan Geschke, Freie Universitšt Berlin

A partial order P has the weak Freese-Nation property (WFN) iff there is a function f : P --> [P ]<w such that for all p, q  (- P with p < q there is r  (- f(p)  /~\ f(q) with p < r < q. For Boolean algebras, the WFN is a generalization of projectivity. The statement ‘P(N) has the WFN’ (WFN(P(N)) for short) is known to be consistent with the axioms of ZFC. More precisely, P(N) has the WFN in every model of set theory obtained by adding a small number (<  ,\' 0) of Cohen reals to a model of CH. (These models will be called Cohen models.) On the other hand, the WFN of P(w) implies a lot of facts about the combinatorics of the reals known to hold in a Cohen model. For example, the WFN on P(N) has a great impact on the values of cardinal invariants of the continuum and on the structure of the automorphism group of P(N)/fin. Therefore it seems save to say that WFN(P(N)) is an axiom capturing the combinatorics (of the reals) in Cohen models.