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Chains and antichains in

Published online by Cambridge University Press:  12 March 2014

James E. Baumgartner
James E. Baumgartner*
Affiliation:
Dartmouth College, Hanover, NH 03755
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Extract

Consider the following propositions:

(A) Every uncountable subset of contains an uncountable chain or antichain (with respect to ⊆).

(B) Every uncountable Boolean algebra contains an uncountable antichain (i.e., an uncountable set of pairwise incomparable elements).

Until quite recently, relatively little was known about these propositions. The oldest result, due to Kunen [4] and the author independently, asserts that if the Continuum Hypothesis (CH) holds, then (A) is false. In fact there is a counter-example 〈Aα: α < ω1〉 such that α < β implies Aβ −Aα is finite. Kunen also observed that Martin's Axiom (MA) + ¬CH implies that no such counterexample 〈Aα: α < ω1〉 exists.

Much later, Komjáth and the author [2] showed that ◊ implies the existence of several kinds of uncountable Boolean algebras with no uncountable chains or antichains. Similar results (but motivated quite differently) were obtained independently by Rubin [5]. Berney [3] showed that CH implies that (B) is false, but his algebra has uncountable chains. Finally, Shelah showed very recently that CH implies the existence of an uncountable Boolean algebra with no uncountable chains or antichains.

Except for Kunen's result cited above, the only result in the other direction was the theorem, due also to Kunen, that MA + ¬CH implies that any uncountable subset of with no uncountable antichains must have both ascending and decending infinite sequences under ⊆.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 1 , March 1980 , pp. 85 - 92
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

[1]Baumgartner, J., All ℵ1-dense sets of reals can be isomorphic, Fundamenta Mathematicae, vol. 79 (1973), pp. 101–106.CrossRefGoogle Scholar
[2]Baumgartner, J. and Komjáth, P., Boolean algebras in which every chain and antichain is countable, Fundamenta Mathematicae (to appear).Google Scholar
[3]Berney, E. S. and Nyikos, P. J., The length and breadth of Boolean algebras (to appear).Google Scholar
[4]Kunen, K., mimeographed notes.Google Scholar
[5]Rubin, M., A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness (to appear).Google Scholar
[6]Solovay, R. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201–245.CrossRefGoogle Scholar