# Martin’s axiom is consistent

If $\kappa $ is an uncountable strong limit cardinal such that for any $$, ${\kappa}^{\lambda}=\kappa $ then it is consistent^{} that ${2}^{{\mathrm{\aleph}}_{0}}=\kappa $ and MA. This is shown by using finite support iterated forcing to construct a model of ZFC in which this is true. Historically, this proof was the motivation for developing iterated forcing.

## Outline

The proof uses the convenient fact that $M{A}_{\kappa}$ holds as long as it holds for all partial orders smaller than $\kappa $. Given the conditions on $\kappa $, there are at most $\kappa $ names for these partial orders. At each step in the forcing^{}, we force with one of these names. The result is that the actual generic subset we add intersects every dense subset of every partial order.

## Construction of ${P}_{\kappa}$

${\widehat{Q}}_{\alpha}$ will be constructed by induction^{} with three conditions: $|{P}_{\alpha}|\le \kappa $ for all $\alpha \le \kappa $, ${\u22a9}_{{P}_{\alpha}}{\widehat{Q}}_{\alpha}\subseteq \mathcal{M}$, and ${P}_{\alpha}$ satisfies the ccc. Note that a partial ordering on a cardinal $$ is a function^{} from $\lambda \times \lambda $ to $\{0,1\}$, so there are at most $$ of them. Since a canonical name for a partial ordering of a cardinal is just a function from ${P}_{\alpha}$ to that cardinal, there are at most ${\kappa}^{{2}^{\lambda}}\le \kappa $ of them.

At each of the $\kappa $ steps, we want to deal with one of these possible partial orderings, so we need to partition^{} the $\kappa $ steps in to $\kappa $ steps for each of the $\kappa $ cardinals less than $\kappa $. In addition, we need to include every ${P}_{\alpha}$ name for any level. Therefore, we partion $\kappa $ into $$ for each cardinal $\delta $, with each ${S}_{\gamma ,\delta}$ having cardinality $\kappa $ and the added condition that $\eta \in {S}_{\gamma ,\delta}$ implies $\eta \ge \gamma $. Then each ${P}_{\gamma}$ name for a partial ordering of $\delta $ is assigned some index $\eta \in {S}_{\gamma ,\delta}$, and that partial order will be dealt with at stage ${Q}_{\eta}$.

Formally, given ${\widehat{Q}}_{\beta}$ for $$, ${P}_{\alpha}$ can be constructed and the ${P}_{\alpha}$ names for partial orderings of each cardinal $\delta $ enumerated by the elements of ${S}_{\alpha ,\delta}$. $\alpha \in {S}_{\gamma ,\delta}$ for some ${\gamma}_{\alpha}$ and ${\delta}_{\alpha}$, and $\alpha \ge {\gamma}_{\alpha}$ so some canonical ${P}_{{\gamma}_{\alpha}}$ name for a partial order ${\widehat{\le}}_{\alpha}$ of ${\delta}_{\alpha}$ has already been assigned to $\alpha $.

Since ${\widehat{\le}}_{\alpha}$ is a ${P}_{{\gamma}_{\alpha}}$ name, it is also a ${P}_{\alpha}$ name, so ${\widehat{Q}}_{\alpha}$ can be defined as $\u27e8{\delta}_{\alpha},{\widehat{\le}}_{\alpha}\u27e9$ if ${\u22a9}_{{P}_{\alpha}}\u27e8{\delta}_{\alpha},{\widehat{\le}}_{\alpha}\u27e9$ satisfies the ccc and by the trivial partial order $\u27e81,\{\u27e81,1\u27e9\}\u27e9$ otherwise. Obviously this satisfies the ccc, and so ${P}_{\alpha +1}$ does as well. Since ${\widehat{Q}}_{\alpha}$ is either trivial or a cardinal together with a canonical name, ${\u22a9}_{{P}_{\alpha}}{\widehat{Q}}_{\alpha}\subseteq \mathcal{M}$. Finally, $|{P}_{\alpha +q}|\le {\sum}_{n}{|\alpha |}^{n}\cdot {({\mathrm{sup}}_{i}|{\widehat{Q}}_{i}|)}^{n}\le \kappa $.

## Proof that $M{A}_{\lambda}$ holds for $$

### Lemma: It suffices to show that $M{A}_{\lambda}$ holds for partial order with size $\le \lambda $

###### Proof.

Suppose $P$ is a partial order with $|P|>\kappa $ and let $$ be dense subsets of $P$. Define functions ${f}_{i}:P\to {D}_{\alpha}$ for $\alpha \kappa $ with ${f}_{\alpha}(p)\ge p$ (obviously such elements exist since ${D}_{\alpha}$ is dense). Let $g:P\times P\to P$ be a function such that $g(p,q)\ge p,q$ whenever $p$ and $q$ are compatible^{}. Then pick some element $q\in P$ and let $Q$ be the closure of $\{q\}$ under ${f}_{\alpha}$ and $g$ with the same ordering as $P$ (restricted to $Q$).

Since there are only $\kappa $ functions being used, it must be that $|Q|\le \kappa $. If $p\in Q$ then ${f}_{\alpha}(p)\ge p$ and clearly ${f}_{\alpha}(p)\in Q\cap {D}_{\alpha}$, so each ${D}_{\alpha}\cap Q$ is dense in $Q$. In addition, $Q$ is ccc: if $A$ is an antichain^{} in $Q$ and ${p}_{1},{p}_{2}\in A$ then ${p}_{1},{p}_{2}$ are incompatible in $Q$. But if they were compatible in $P$ then $g({p}_{1},{p}_{2})\ge {p}_{1},{p}_{2}$ would be an element of $Q$, so they must be incompatible in $P$. Therefore $A$ is an antichain in $P$, and therefore must have countable^{} cardinality, since $P$ satisfies the ccc.

By assumption^{}, there is a directed $G\subseteq Q$ such that $G\cap ({D}_{\alpha}\cap Q)\ne \mathrm{\varnothing}$ for each $$, and therefore $M{A}_{\lambda}$ holds in full.
∎

Now we must prove that, if $G$ is a generic subset of ${P}_{\kappa}$, $R$ some partial order with $|R|\le \lambda $ and $$ are dense subsets of $R$ then there is some directed subset of $R$ intersecting each ${D}_{\alpha}$.

If $$ then $\lambda $ additional elements can be added greater than any other element of $R$ to make $|R|=\lambda $, and then since there is an order isomorphism into some partial order of $\lambda $, assume $R$ is a partial ordering of $\lambda $. Then let $D=\{\u27e8\alpha ,\beta \u27e9\mid \alpha \in {D}_{\beta}\}$.

Take canonical names so that $R=\widehat{R}[G]$, $D=\widehat{D}[G]$ and ${D}_{i}={\widehat{D}}_{i}[G]$ for each $$ and:

$$\begin{array}{cc}\hfill {\u22a9}_{{P}_{\kappa}}& \widehat{R}\mathtt{\text{is a partial ordering satisfying ccc and}}\hfill \\ & \widehat{D}\subseteq \lambda \times \lambda \mathtt{\text{and}}\hfill \\ & \widehat{{D}_{\alpha}}\mathtt{\text{is dense in}}\widehat{R}\hfill \end{array}$$ |

For any $\alpha ,\beta $ there is a maximal antichain ${D}_{\alpha ,\beta}\subseteq {P}_{\kappa}$ such that if $p\in {D}_{\alpha ,\beta}$ then either $p{\u22a9}_{{P}_{\kappa}}\alpha {\le}_{\widehat{R}}\beta $ or $p{\u22a9}_{{P}_{\kappa}}\alpha {\nleqq}_{\widehat{R}}\beta $ and another maximal antichain ${E}_{\alpha ,\beta}\subseteq {P}_{\kappa}$ such that if $p\in {E}_{\alpha ,\beta}$ then either $p{\u22a9}_{{P}_{\kappa}}\u27e8\alpha ,\beta \u27e9\in \widehat{D}$ or $p{\u22a9}_{{P}_{\kappa}}\u27e8\alpha ,\beta \u27e9\notin \widehat{D}$. These antichains determine the value of those two formulas^{}.

Then, since ${\kappa}^{\mathrm{cf}\kappa}>\kappa $ and ${\kappa}^{\mu}=\kappa $ for $$, it must be that $\mathrm{cf}\kappa =\kappa $, so $\kappa $ is regular^{}. Then $$, so ${D}_{\alpha ,\beta},{E}_{\alpha ,\beta}\subseteq {P}_{\gamma}$, and therefore the ${P}_{\kappa}$ names $\widehat{R}$ and $\widehat{D}$ are also ${P}_{\gamma}$ names.

### Lemma: For any $\gamma $, ${G}_{\gamma}=\{p\upharpoonright \gamma \mid p\in G\}$ is a generic subset of ${P}_{\gamma}$

###### Proof.

First, it is directed, since if ${p}_{1}\upharpoonright \gamma ,{p}_{2}\upharpoonright \gamma \in {G}_{\gamma}$ then there is some $p\in G$ such that $p\le {p}_{1},{p}_{2}$, and therefore $p\upharpoonright \gamma \in {G}_{\gamma}$ and $p\le {p}_{1}\upharpoonright \gamma ,{p}_{2}\upharpoonright \gamma $.

Also, it is generic. If $D$ is a dense subset of ${P}_{\gamma}$ then ${D}_{\kappa}=\{p\in {P}_{\kappa}\mid p\le q\in D\}$ is dense in ${P}_{\kappa}$, since if $p\in {P}_{\kappa}$ then there is some $d\le p\upharpoonright $, but then $d$ is compatible with $p$, so $d\cup p\in {D}_{\kappa}$. Therefore there is some $p\in {D}_{\kappa}\cap {G}_{\kappa}$, and so $p\upharpoonright \in D\cap {G}_{\gamma}$. ∎

Since $\widehat{R}$ and $\widehat{D}$ are ${P}_{\gamma}$ names, $\widehat{R}[G]=\widehat{R}[{G}_{\gamma}]=R$ and $\widehat{D}[G]=\widehat{D}[{G}_{\gamma}]=D$, so

$$\begin{array}{cc}\hfill V[{G}_{\gamma}]\models & \widehat{R}\mathtt{\text{is a partial ordering of}}\lambda \mathtt{\text{satisfying the ccc and}}\hfill \\ & \widehat{{D}_{\alpha}}\mathtt{\text{is dense in}}\widehat{R}\hfill \end{array}$$ |

Then there must be some $p\in {G}_{\gamma}$ such that

$$p{\u22a9}_{{P}_{\gamma}}\widehat{R}\mathtt{\text{is a partial ordering of}}\lambda \mathtt{\text{satisfying the ccc}}$$ |

Let ${A}_{p}$ be a maximal antichain of ${P}_{\gamma}$ such that $p\in {A}_{p}$, and define ${\widehat{\le}}^{*}$ as a ${P}_{\gamma}$ name with $(p,m)\in {\widehat{\le}}^{*}$ for each $m\in \widehat{R}$ and $(a,n)\in {\widehat{\le}}^{*}$ if $n=(\alpha ,\beta )$ where $$ and $p\ne a\in {A}_{p}$. That is, ${\widehat{\le}}^{*}[G]=R$ when $p\in G$ and ${\widehat{\le}}^{*}[G]=\in \upharpoonright \lambda $ otherwise. Then this is the name for a partial ordering of $\lambda $, and therefore there is some $\eta \in {S}_{\gamma ,\lambda}$ such that ${\widehat{\le}}^{*}={\widehat{\le}}_{\eta}$, and $\eta \ge \gamma $. Since $p\in {G}_{\gamma}\subseteq {G}_{\eta}$, ${\widehat{Q}}_{\eta}[{G}_{\eta}]={\widehat{\le}}_{\eta}[{G}_{\eta}]=R$.

Since ${P}_{\eta +1}={P}_{\eta}*{Q}_{\eta}$, we know that ${G}_{{Q}_{\eta}}\subseteq {Q}_{\eta}$ is generic since http://planetmath.org/node/3258forcing with the composition^{} is equivalent^{} to successive forcing. Since ${D}_{i}\in V[{G}_{\gamma}]\subseteq V[{G}_{\eta}]$ and is dense, it follows that ${D}_{i}\cap {G}_{{Q}_{\eta}}\ne \mathrm{\varnothing}$ and since ${G}_{{Q}_{\eta}}$ is a subset of $R$ in ${P}_{\kappa}$, $M{A}_{\lambda}$ holds.

## Proof that ${2}^{{\mathrm{\aleph}}_{0}}=\kappa $

The relationship between Martin’s axiom and the continuum hypothesis^{} tells us that ${2}^{{\mathrm{\aleph}}_{0}}\ge \kappa $. Since ${2}^{{\mathrm{\aleph}}_{0}}$ was less than $\kappa $ in $V$, and since $|{P}_{\kappa}|=\kappa $ adds at most $\kappa $ elements, it must be that ${2}^{{\mathrm{\aleph}}_{0}}=\kappa $.

Title | Martin’s axiom is consistent |
---|---|

Canonical name | MartinsAxiomIsConsistent |

Date of creation | 2013-03-22 13:21:59 |

Last modified on | 2013-03-22 13:21:59 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Result |

Classification | msc 03E50 |