# a shorter proof: Martin’s axiom and the continuum hypothesis

This is another, shorter, proof for the fact that $M{A}_{{\mathrm{\aleph}}_{0}}$ always holds.

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

Let $(P,\le )$ 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 ${D}_{1},\mathrm{\dots},{D}_{n},\mathrm{\dots}$ be the dense subsets in $\mathcal{D}$. Furthermore let ${p}_{0}=p$. Now we can choose for every $$ an element ${p}_{n}\in P$ such that ${p}_{n}\le {p}_{n-1}$ and ${p}_{n}\in {D}_{n}$. If we now consider the set $$, 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.
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Title | a shorter proof: Martin’s axiom and the continuum hypothesis^{} |
---|---|

Canonical name | AShorterProofMartinsAxiomAndTheContinuumHypothesis |

Date of creation | 2013-03-22 13:53:58 |

Last modified on | 2013-03-22 13:53:58 |

Owner | x_bas (2940) |

Last modified by | x_bas (2940) |

Numerical id | 11 |

Author | x_bas (2940) |

Entry type | Proof |

Classification | msc 03E50 |

Defines | $\mathcal{D}$-generic |

Defines | generic |

Defines | dense |