On Ordinals Accessible by Infinitary Languages

by Shelah and Vaisanen and Vaananen. [ShVV:812]
Fundamenta Math, 2005
Let lambda be an infinite cardinal number. The ordinal number delta (lambda) is the least ordinal gamma such that if phi is any sentence of L_{lambda^+ omega}, with a unary predicate D and a binary predicate prec, and phi has a model M with < D^M, prec^M> a well-ordering of type >= gamma, then phi has a model M' where < D^{M'}, prec^{M'}> is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L_{lambda^+ omega} is exactly beth_{delta (lambda)} . We show the following theorem. Theorem Suppose aleph_0< lambda < theta <= kappa are cardinal numbers such that lambda^{< lambda}= lambda, cf (theta) >= lambda^+ and mu^lambda < theta whenever mu < theta, and kappa^lambda = kappa . Then there is a forcing extension preserving all cofinalities, adding no new sets of cardinality < lambda such that in the extension 2^lambda = kappa and delta (lambda)= theta .

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