@MISC{Cummings_souslintrees, author = {James Cummings}, title = {Souslin Trees Which Are Hard To Specialise}, year = {} }

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Abstract

. We construct some + -Souslin trees which cannot be specialised by any forcing which preserves cardinals and cofinalities. For a regular cardinal we use the principle, for singular we use squares and diamonds. 1. Introduction We start by recalling a few basic definitions concerning trees. For more information, see [4]. Definition 1.1. Let be a regular cardinal. 1. (T ; ! T ) is a tree iff ! T is a partial ordering of T such that f y : y ! T x g is well ordered by ! T for all x 2 T . 2. Let T be a tree. Then (a) If x 2 T , ht T (x) is the order type of (f y : y ! T x g; ! T ). (b) T ff = f x 2 T : ht T (x) = ff g. (c) ht(T ) is the least ff such that T ff = ;. (d) T ff = S fi!ff T fi . (e) A cofinal branch of T is a set B ` T such that B is linearly ordered by ! T , and 8ff ! ht(T ) 9b 2 B ht T (b) ff. 3. T is a -tree iff ht(T ) = and jT ff j ! for all ff ! . 4. T is a -Aronszajn tree iff T is a -tree with no cofinal branch. 5. T is a special + -tree iff ...