### 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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