Theorems · Definition · logic and foundations
Ordinal.omega
Ordinal.{u_1} ↪o Ordinal.{u_1}The omega function gives the infinite initial ordinals listed by their ordinal index.
omega 0 = ω, omega 1 = ω₁ is the first uncountable ordinal, and so on.
This is not to be confused with the first infinite ordinal Ordinal.omega0.
For a version including finite ordinals, see Ordinal.preOmega.
Conventions for notations in identifiers:
* The recommended spelling of ω_ in identifiers is omega.
- Defined in
- Mathlib.SetTheory.Cardinal.Aleph
- Cited by
- 60 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Ordinalstatement · cited by 1,688
- OrderEmbeddingstatement · cited by 619
- Ordinal.omega0proof · cited by 197
- Ordinal.preOmegaproof · cited by 31
- RelEmbedding.transproof · cited by 27
- OrderEmbedding.addLeftproof · cited by 5
Cited by60
Results whose statement or proof uses this declaration.
- Ordinal.lift_omegastatement · cited by 12
- Cardinal.ord_alephstatement · cited by 7
- Ordinal.omega0_le_omegastatement and proof · cited by 4
- Ordinal.isInitial_omegastatement · cited by 4
- Ordinal.omega_eq_preOmegastatement · cited by 3
- Ordinal.omega_zerostatement · cited by 3
- MeasurableSpace.cardinal_generateMeasurable_leproof · cited by 2
- Cardinal.isSuccLimit_omegastatement · cited by 2
- MeasurableSpace.generateMeasurable_eq_recstatement and proof · cited by 2
- Ordinal.IsInitial.isPrincipal_opowproof · cited by 2
- Cardinal.cof_omega_onestatement · cited by 2
- Ordinal.isPrincipal_opow_omegastatement and proof · cited by 2