Martin's Axiom

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In the mathematical field of set theory, **Martin's axiom**, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH. Indeed, it is only really of interest when the continuum hypothesis fails (otherwise it adds nothing to ZFC). It can informally be considered to say that all cardinals less than the cardinality of the continuum, <math></math>, behave roughly like <math>aleph_0</math>. The intuition behind this can be understood by studying the proof of the Rasiowa-Sikorski lemma. More formally it is a principle that is used to control certain forcing arguments.

## Statement of Martin's axiom

The various statements of Martin's axiom typically take two parts. MA(**k**) says that for any partial order <math>P</math> satisfying the countable chain condition (hereafter **ccc**) and any family <math>D</math> of dense sets in <math>P</math>, with <math>|D|</math> at most **k**, there is a filter <math>F</math> on <math>P</math> such that <math>F</math> ∩ <math>d</math> is non-empty for every <math>d in D</math>. MA, then, says that MA(**k**) holds for every **k** less than the continuum. (It is a theorem of ZFC that MA(<math></math>) fails.) Note that, in this case (for...

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