Delta lemma
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The Δ-lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collection of pairwise incompatible elements in a forcing poset. It may for example be used as one of the ingredients in a proof showing that it is consistent with ZFC that the continuum hypothesis does not hold.
[edit] Formal definition
A Δ-system W is a collection of sets whose pairwise intersection is constant. That is, there exists a fixed S (possibly empty) such that for all A, B ∈ W with A ≠ B, A ∩ B = S.
The Δ-lemma states that every uncountable collection of finite sets contains an uncountable Δ-system.
[edit] References
- Jech, Thomas (2003). Set Theory. Springer.
- Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0.
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