Universal Complete Boolean Algebras and Cardinal Collapsing

✍ Scribed by J. L. Bell

• Publisher: John Wiley and Sons
• Year: 1976
• Tongue: English
• Weight: 192 KB
• Volume: 22
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19760220121

✦ Synopsis

Lct 1 be an infinite cardinal and let A , B be Boolean algebras. A homomorphism h: , 4 4 B is said to be A-cmpkte if whenever X is a subset of A of cardinality I such that the join V X of X exists in A , then V h[X] exists in B and is equal to h(V X ) .

If x is an infinite cardinal, B is said to be ( x , il)-universaZ if for each Boolean algebra A of cardinality ~t there is a A-complete monomorphism (i.e. one-one homomorphism) of A into B.

Our objective in this paper is to investigate, for compZete Boolean algebras B, the relationship between (x, >)-universality of B and the collapsing of cardinals to countable ordinals in the B-extension of the universe of sets. We assume familiarity with the technique of Boolean-valued models of set theory, as presented, e.g., in JECH [l].

For the theory of Boolean algebras, see SIKHORSKI

For each set X , we write 1 x 1 for the cardinality of X . The symbols 5, y will always denote ordinals, while a , p, x , 1 will be used for cardinals. If A is a Boolean algebra.

we write 0, and lA for the least and greatest elements of A , respectively. If x, y E A , then x*, x v y, z A y denote the complement of x, and the join and meet of x and y, respectively. A subset X of A is called an antichain in A if OA 4 X and x A y = 0 I