A universePlanetmathPlanetmath 𝐔 is a nonempty set satisfying the following axioms:

  1. 1.

    If xβˆˆπ” and y∈x, then yβˆˆπ”.

  2. 2.

    If x,yβˆˆπ”, then {x,y}βˆˆπ”.

  3. 3.

    If xβˆˆπ”, then the power setMathworldPlanetmath 𝒫⁒(x)βˆˆπ”.

  4. 4.

    If {xi|i∈Iβˆˆπ”} is a family of elements of 𝐔, then βˆͺi∈Ixiβˆˆπ”.

From these axioms, one can deduce the following properties:

  1. 1.

    If xβˆˆπ”, then {x}βˆˆπ”.

  2. 2.

    If x is a subset of yβˆˆπ”, then xβˆˆπ”.

  3. 3.

    If x,yβˆˆπ”, then the ordered pairMathworldPlanetmath (x,y)={{x,y},x} is in 𝐔.

  4. 4.

    If x,yβˆˆπ”, then xβˆͺy and xΓ—y are in 𝐔.

  5. 5.

    If {xi|i∈Iβˆˆπ”} is a family of elements of 𝐔, then the productPlanetmathPlanetmath ∏i∈Ixi is in 𝐔.

  6. 6.

    If xβˆˆπ”, then the cardinality of x is strictly less than the cardinality of 𝐔. In particular, π”βˆ‰π”.

In order for uncountable universes to exist, it is necessary to adopt an extra axiom for set theoryMathworldPlanetmath. This is usually phrased as:

Axiom 1.

For every cardinal Ξ±, there exists a strongly inaccessible cardinal Ξ²>Ξ±.

This axiom cannot be proven using the axioms ZFC. But it seems (according to Bourbaki) that it probably cannot be proven not to lead to a contradictionMathworldPlanetmathPlanetmath.

One usually also assumes

Axiom 2.

For every set X, there is no infinite descending chain β‹―βˆˆx2∈x1∈X; this is called being artinian.

This axiom does not affect the consistency of ZFC, that is, ZFC is consistent if and only if ZFC with this axiom added is consistent. This is also known as the axiom of foundation, and it is often included with ZFC. If it is not accepted, then one can for all practical purposes restrict oneself to working within the class of artinian sets.

Finally, one must be careful when using relationsMathworldPlanetmath within universes; the details are too technical for Bourbaki to work out (!), but see the appendix to ExposΓ© 1 of [SGA4] for more detail.

The standard reference for universes is [SGA4].


  • SGA4 Grothendieck et al. Seminaires en Geometrie Algebrique 4, Tome 1, ExposΓ© 1 (or the appendix to ExposΓ© 1, by N. Bourbaki for more detail and a large number of results there described as β€œne pouvant servir Γ  rien”). SGA4 is http://www.math.mcgill.ca/Β archibal/SGA/SGA.htmlavailable on the Web. (It is in French.)
Title universe
Canonical name Universe
Date of creation 2013-03-22 13:31:13
Last modified on 2013-03-22 13:31:13
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 8
Author archibal (4430)
Entry type Definition
Classification msc 03E30
Classification msc 18A15
Related topic Small