---
tag: V8KGT8MRFE
kind: concept
name: Martin's axiom
msc: ["03E50", "03E57"]
sources:
  - url: https://en.wikipedia.org/wiki/Martin's_axiom
same_as:
  wikidata: Q622185
  wikipedia: Martin's axiom
---

## Summary

Martin's axiom, MA, is about partial orders with the countable chain condition.
It says that one filter can meet any fewer than continuum many dense sets of such an order.
It is a statement about [[forcing|F2DTWW9HP7]] that can be used without forcing, and the [[precise definition|Z8XMCCZFA4]] is given separately.
It follows from the [[continuum hypothesis|0MN2MER9WN]], but it is also consistent with its failure.
In that case it decides many questions that ZFC leaves open, for instance [[Suslin's problem|D1T39KHPDN]].

## Explanations

### L1 human

Many constructions in mathematics build an object step by step, meeting countably many requirements one after another.
The Baire category theorem says that such constructions work when there are countably many requirements.
Martin's axiom says they still work with more requirements, as long as there are fewer than the number of real numbers and the steps satisfy a mild compatibility condition.
For example, it implies that the union of fewer than continuum many meagre subsets of the real line is meagre.