Definition:Universe (Set Theory)
This page is about Universe in the context of Set Theory. For other uses, see Universe.
Definition
Sets are considered to be subsets of some large universal set, also called the universe.
Exactly what this universe is will vary depending on the subject and context.
When discussing particular sets, it should be made clear just what that universe is.
However, note that from There Exists No Universal Set, this universe cannot be everything that there is.
The traditional symbol used to signify the universe is $\mathfrak A$.
However, this is old-fashioned and inconvenient, so some newer texts have taken to using $\mathbb U$ or just $U$ instead.
With this notation, this definition can be put into symbols as:
- $\forall S: S \subseteq \mathbb U$
The use of $\mathbb U$ or a variant is not universal: some sources use $X$.
Zermelo-Fraenkel Theory
If the universal class is allowed to be a set in ZF(C) set theory, then a contradiction results. One equivalent of the axiom of specification states that:
- $\forall z: \forall A: \paren {A \subseteq z \implies A \in U}$
However, we may conservatively extend the ZFC axioms to incorporate classes, which is done in von Neumann–Bernays–Gödel (NBG) set theory.
The basic gadget we work with is a class, and a set is defined to be an element of a class. Some refer to a proper class to be one which is not contained in any class (so an "improper" class would be one that is contained in some class, thus it is a set).
We avoid the contradiction mentioned by modifying the axiom of specification through restricting quantifiers to range over sets, but not all classes. (We also demand that sets are not bijective to the class of all ordinals.)
Michael Shulman's "Set theory for category theory" (arXiv:0810.1279v2 [math.CT]) studies various esoteric foundational issues relevant for category theory, and gives the definition of a Universe in any set theory as: a model for the ZFC axioms. This covers the von Neumann universe, the Grothendieck universe, the class of all sets in NBG set theory, etc.
However, some alternative set theories, such as Quine's New Foundations, allow the universal set to be a value of a variable, and reject certain instances of the axiom of specification.
All the elements of the universal set are precisely the Universe of Discourse of quantification.
Also known as
Some sources refer to the universal set as the universe of discourse, which name is also used for a similar concept in the category of predicate logic.
Some sources (particularly in the context of probability theory) refer to it as the space.
Also see
- Results about the universal set can be found here.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 2$: The Axiom of Specification
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Set Theory: $1.2$: Sets and subsets
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Set-Theoretic Notation
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B ii}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \beta$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.5$: Complementation
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.1$: Sets and Subsets
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6$: Subsets
- 1981: G. de Barra: Measure Theory and Integration ... (next): Chapter $1$: Preliminaries: $1.1$ Set Theory
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: universal set or universe
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: universe: 1.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: universal set
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: universal set
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: universal set
As such, the following source works, along with any process flow, will need to be reviewed.
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- 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts