THE SPECTRUM OF THE GAMMA-INVARIANT OF A BILINEAR SPACE

To every symmetric bilinear space (X, phi) of regular uncountable dime nsion kappa, an invariant Gamma(X, phi) is an element of P(kappa)/F(ka ppa) (where F(kappa) is the club filter) can be assigned. We prove tha t in dimension aleph(2) the spectrum of Gamma cannot be determined in ZFC. For this, on the one hand we show that under CH, Gamma attains th e maximal (with respect to a restriction provable in ZFC) spectrum; we also show that CH is not necessary for this result. On the other hand we show that in a variation of Mitchell's model, which is obtained by collapsing a weakly compact cardinal to omega(2) the spectrum of Gamm a in dimension aleph(2) is much thinner than the maximal one. (C) 1997 Academic Press.