en>David Eppstein: :Category:Geometric dissection

2012-08-06T01:29:09Z

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In [[set theory]], a '''nice name''' is a concept used in [[forcing (mathematics)|forcing]] to impose an upper bound on the number of subsets in the generic model. It is a technical concept used in the context of forcing to prove independence results in set theory such as [[Easton's theorem]].

==Formal definition==
Let <math>M \models</math> ZFC be transitive, <math>(\mathbb{P}, <)</math> a forcing notion in <math>M</math>, and suppose <math>G \subseteq \mathbb{P}</math> is generic over <math>M</math>. Then for any <math>\mathbb{P}</math>-[[Set theory|name]] in <math>M</math>, <math>\tau</math>,

<math>\eta</math> is a nice name for a subset of <math>\tau</math> if <math>\eta</math> is a <math>\mathbb{P}</math>-name satisfying the following properties:

(1) <math>\textrm{dom}(\eta) \subseteq \textrm{dom}(\tau)</math>

(2) For all <math>\mathbb{P}</math>-names <math>\sigma \in M</math>, <math>\{p \in \mathbb{P}| \langle\sigma, p\rangle \in \eta\}</math> forms an antichain.

(3) '''(Natural addition)''': If <math>\langle\sigma, p\rangle \in \eta</math>, then there exists <math>q \geq p</math> in <math>\mathbb{P}</math> such that <math>\langle\sigma, q\rangle \in \tau</math>.

==References==
* Kenneth Kunen (1980) ''Set theory: an introduction to independence proofs'', Volume 102 of Studies in logic and the foundations of mathematics (Elsevier) ISBN 0-444-85401-0, p.208

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[[Category:Forcing (mathematics)]]
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en>David Eppstein: :Category:Geometric dissection