---
tag: HFQSSB1Z23
kind: concept
name: Diamond principle
msc: ["03E65", "03E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Diamond_principle
same_as:
  wikidata: Q1734160
  wikipedia: Diamond principle
---

## Summary

The diamond principle, written $\diamondsuit$, is a combinatorial axiom about $\omega_1$ introduced by Ronald Jensen.
It says that a sequence of guesses, one at each countable ordinal, correctly predicts every subset of $\omega_1$ on a [[stationary set|YE6WZR46RQ]] of ordinals.
It holds in the [[constructible universe|6TF2JBTPPJ]], and it implies the [[continuum hypothesis|0MN2MER9WN]].
It is used to build objects such as Suslin trees, and the [[precise definition|J8GYFAA70H]] is given separately.

## Explanations

### L1 human

Imagine an oracle that, at each countable stage $\alpha$, makes one guess $A_\alpha$ about what a hidden set $A \subseteq \omega_1$ looks like below $\alpha$.
Diamond says that there is an oracle so good that, for every possible hidden set, its guesses are right at a large set of stages.
This is surprising, because there are $2^{\aleph_1}$ subsets of $\omega_1$, while each stage gets only one guess.
Such an oracle lets one kill every possible obstacle during a construction of length $\omega_1$, and that is how it is used.