Martin's axiom

Martin's axiom is a statement in set theory that can not be proven or refuted in the usual Zermelo-Fraenkel system . It was introduced in 1970 by Donald A. Martin and Robert M. Solovay .

Write for the cardinality of the continuum. ${\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}$

Be a partial order . A lot means tight here when a lot of the shape meets. Let be a set of dense subsets of . We are looking for a filter on that hits all elements , i. H. does not cut empty ; is then called -generic filter. The Rasiowa-Sikorski lemma says that it is always possible for countable ones to find such a filter . The situation is different for uncountable sets : If ${\displaystyle \langle P,\leq _{P}\rangle }$ ${\displaystyle D\subseteq P}$ ${\displaystyle D}$ ${\displaystyle P^{\leq p}=\{q\in P\mid q\leq p\}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle P}$ ${\displaystyle F}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle F}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle F}$ ${\displaystyle {\mathcal {D}}}$

• ${\displaystyle \left|{\mathcal {D}}\right|={\mathfrak {c}}}$ or
• ${\displaystyle \langle P,\leq _{P}\rangle }$has uncountable anti - chains ,

there are generally no -generic filters. ${\displaystyle {\mathcal {D}}}$

Martin's axiom for an infinite cardinal number , in short , is the statement, ${\displaystyle \kappa }$ ${\displaystyle \mathrm {MA} (\kappa )}$

for every partial order that has only countable anti-chains and any set of dense subsets with there is a -generic filter . ${\displaystyle \langle P,\leq _{P}\rangle }$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \left|{\mathcal {D}}\right|<\kappa }$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle F}$

Thus the statement of the lemma of Rasiowa-Sikorski is provably true and exact. For is provably wrong. If the continuum hypothesis (CH) applies , the statement is therefore decided for every infinite cardinal number. Consequently, Martin's axiom for an infinite cardinal number is only of interest in models in which the continuum hypothesis fails. ${\displaystyle \mathrm {MA} (\aleph _{0})}$ ${\displaystyle \kappa \geq {\mathfrak {c}}}$ ${\displaystyle \mathrm {MA} (\kappa )}$

The term Martin's axiom without specification of a cardinal number, in short , denotes the statement that ${\displaystyle \mathrm {MA} }$

${\displaystyle \mathrm {MA} (\kappa )}$holds for all uncountable . ${\displaystyle \kappa <{\mathfrak {c}}}$

The formulation "[...] for all infinite " is equivalent , since the difference only affects. Martin's axiom without the formulation for an infinite cardinal number is equivalent to that it ${\displaystyle \kappa <{\mathfrak {c}}}$ ${\displaystyle \mathrm {MA} (\aleph _{0})}$

for every partial order that has only countable anti-chains and any set of dense subsets with a -generic filter . ${\displaystyle \langle P,\leq _{P}\rangle }$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \left|{\mathcal {D}}\right|<{\mathfrak {c}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle F}$

Provided that ZFC is consistent, models of ZFC + MA + ¬CH can be constructed, i.e. in which Martin's axiom applies but the continuum hypothesis does not.

Martin's axiom clearly means that the uncountable cardinal numbers are in a certain sense small compared to and behave similarly . ${\displaystyle \kappa <{\mathfrak {c}}}$ ${\displaystyle {\mathfrak {c}}}$ ${\displaystyle \aleph _{0}}$

literature

• Jech, Thomas: Set Theory , Springer-Verlag Berlin Heidelberg (2006), ISBN 3-540-44085-2 .
• Kunen, Keneth: Set Theory: An Introduction to Independence Proofs , North-Holland (1980), ISBN 0-444-85401-0 .
• Martin, DA; Solovay, RM: Internal Cohen extensions , Ann. Math. Logic 2 (2) (1970): 143-178