Δ lemma

The -Lemma is a mathematical theorem from combinatorial set theory . It is used in the development of the forcing method. ${\displaystyle \Delta }$

statement

Be a family of sets, and another set. is called a system with a root if: ${\displaystyle D}$ ${\displaystyle d}$ ${\displaystyle D}$ ${\displaystyle \Delta }$ ${\displaystyle d}$

• ${\displaystyle \forall x\neq y\in D:x\cap y=d}$, the intersection of two sets is constant. ${\displaystyle D}$

The -Lemma says: Every uncountable family of finite sets contains an uncountable system. ${\displaystyle \Delta }$ ${\displaystyle \Delta }$

generalization

The lemma can be generalized as follows: Let cardinal numbers with ${\displaystyle \lambda <\mu }$

• ${\displaystyle \mu }$is regular : ${\displaystyle \mu =cf(\mu )}$
• The following applies to all : (see cardinal number arithmetic ), ${\displaystyle \alpha <\mu }$ ${\displaystyle \alpha ^{<\lambda }:=\sup _{\gamma <\lambda }\alpha ^{\gamma }<\mu }$

then there is for every family with and for a system of power . If one sets and , one obtains the above special case. ${\displaystyle I}$ ${\displaystyle \left|I\right|=\mu }$ ${\displaystyle \left|a\right|<\lambda }$ ${\displaystyle a\in I}$ ${\displaystyle \Delta }$ ${\displaystyle \mu }$ ${\displaystyle \lambda =\aleph _{0}}$ ${\displaystyle \mu =\aleph _{1}}$

literature

• Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded, corrected 4th print. Springer, Berlin et al. 2006, ISBN 3-540-44085-2 .
• Kenneth Kunen : Set Theory. An Introduction to Independence Proofs (= Studies in Logic and the Foundations of Mathematics. Vol. 102). North-Holland Publishing Co., Amsterdam et al. 1980, ISBN 0-444-85401-0 .