Collapsing algebra: Difference between revisions

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In mathematics, a '''collapsing algebra''' is a type of [[Boolean algebra (structure)|Boolean algebra]] sometimes used in [[Forcing (mathematics)|forcing]] to reduce ("collapse") the size of [[Cardinal number|cardinals]]. The [[poset]]s used to generate collapsing algebras were introduced by {{harvs|txt|first=[[Azriel|last= Lévy]] in 1963.{{sfn|authorlink=Azriel Lévy|year1963|p=1963593}}.

The collapsing algebra of λ^ω is a [[complete Boolean algebra]] with at least λ elements but generated by a countable number of elements. As the size of countably generated complete Boolean algebras is unbounded, this shows that there is no [[Free algebra|free]] complete Boolean algebra on a countable number of elements.

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==References==

{{reflist}}

* {{cite book | last=Bell | first=J. L. | year=1985 | title=Boolean-Valued Models and Independence Proofs in Set Theory | edition=2nd | ___location=Oxford | publisher=Oxford University Press (Clarendon Press) | series=Oxford Logic Guides | volume=12 | isbn=0-19-853241-5 | zbl=0585.03021 | url-access=registration | url=https://archive.org/details/booleanvaluedmod0000bell }}

* {{cite book | authorlinkauthor-link=Thomas Jech|last=Jech | first=Thomas| title=Set theory | edition=third millennium (revised and expanded) | publisher=[[Springer-Verlag]] | year=2003 | isbn=3-540-44085-2 | oclc=174929965 | zbl=1007.03002}}

* {{cite journal | last=Lévy | first=Azriel |author-link=Azriel Lévy |title=Independence results in set theory by Cohen's method. IV, | journal=Notices Amer. Math. Soc. |volume=10 |year=1963|page= 593}}

[[Category:Boolean algebra]]

[[Category:Forcing (mathematics)]]

{{algebra-stub}}