---
tag: PEXTM3MJ5Z
kind: concept
name: Axiom of determinacy
msc: ["03E60", "03E25"]
sources:
  - url: https://en.wikipedia.org/wiki/Axiom_of_determinacy
same_as:
  wikidata: Q4830558
  wikipedia: Axiom of determinacy
---

## Summary

The axiom of determinacy, written AD, says that every infinite game of natural numbers with perfect information is [[determined|QXECDPAYTN]].
Jan Mycielski and Hugo Steinhaus proposed it in 1962 as an alternative to the [[axiom of choice|0HTGB13FE2]], which it contradicts.
In ZF together with AD, every set of real numbers is Lebesgue measurable, has the [[Baire property|FZ7237VY65]] and has the [[perfect set property|H47ZGRR13V]].
AD is studied in models such as $L(\R)$, where it can hold under large cardinal hypotheses.

## Explanations

### L1 human

Pick any rule that sorts infinite sequences of natural numbers into wins and losses for the first player.
The axiom says that, whatever the rule, one of the two players has a strategy that always wins.
With the axiom of choice one can write down a strange rule for which no player has such a strategy, so AD and choice cannot both hold.
In exchange, AD removes the pathological sets of reals that choice creates, such as non-measurable sets.