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[parent] a shorter proof: Martin's axiom and the continuum hypothesis (Proof)

This is another, shorter, proof for the fact that $ MA_{\aleph_0}$ always holds.

Let $ (P,\leq)$ be a partially ordered set and $ \mathcal D$ be a collection of subsets of $ P$. We remember that a filter $ G$ on $ (P,\leq)$ is $ \mathcal D$-generic if $ G \cap D \neq \varnothing$ for all $ D \in \mathcal D$ which are dense in $ (P,\leq)$. (In this context “dense” means: If $ D$ is dense in $ (P,\leq)$, then for every $ p \in P$ there's a $ d \in D$ such that $ d \leq p$.)

Let $ (P,\leq)$ be a partially ordered set and $ \mathcal D$ a countable collection of dense subsets of $ P$. Then there exists a $ \mathcal D$-generic filter $ G$ on $ P$. Moreover, it could be shown that for every $ p \in P$ there's such a $ \mathcal D$-generic filter $ G$ with $ p \in G$.

Proof. Let $ D_1,\dots, D_n, \dots$ be the dense subsets in $ \mathcal D$. Furthermore let $ p_0 = p$. Now we can choose for every $ 1 \leq n < \omega$ an element $ p_n \in P$ such that $ p_n \leq p_{n-1}$ and $ p_n \in D_n$. If we now consider the set $ G:=\{ q \in P \mid \exists \; n < \omega$    s.t. $ p_n \leq q \}$, then it is easy to check that $ G$ is a $ \mathcal D$-generic filter on $ P$ and $ p \in G$ obviously. This completes the proof. $ \qedsymbol$



"a shorter proof: Martin's axiom and the continuum hypothesis" is owned by x_bas. [ full author list (2) | owner history (1) ]
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Also defines:  $\mathcal D$-generic, generic, dense
Keywords:  Martins Axiom, MA_{\aleph_0} countable collections of dense subsets, internal forcing theorem

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Cross-references: dense subsets, countable, dense in, filter, subsets, collection, partially ordered set
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This is version 8 of a shorter proof: Martin's axiom and the continuum hypothesis, born on 2003-08-24, modified 2004-03-15.
Object id is 4647, canonical name is SomethingRelatedToMartinsAxiomAndTheContinuumHypothesis.
Accessed 11834 times total.

Classification:
AMS MSC: 03E50 (Mathematical logic and foundations :: Set theory :: Continuum hypothesis and Martin's axiom)
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