Definition:Universe (Set Theory)
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:
- $\forall S: S \subseteq \mathbb U$
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 Subsets states that:
- $\forall z: \forall A: \paren {A \subseteq z \implies A \in U}$
We have not gone very far down the route of "Class Theory" yet (if we've started it at all) so I'm not sure how much background work there is to distinguish between "universal class" as mentioned above and "universal set" which is the subject of this page. If in fact the relevant definition will naturally lead on to the discussion as raised here, then we might need to move this into such a page. So we might need to put some words in here to talk about that, but might be worth while waiting till we've covered class theory. Till then this paragraph can stay, as it's useful.
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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 Subsets 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 Subsets.
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All the elements of the universal set are precisely the Universe of Discourse of quantification.
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Work In Progress: Discussion of the topics in some or more of the above dissertation is already under way in the various pages to which it is more immediately relevant. In cases where there is not, there are plans in place (my old standby: "I haven't got round to that bit yet") - in particular there is a plan to include NBG as a series of entries (however much it takes) so as to lay the axiomatic framework for individual classes. I suggest that we might want to put the above discussion in the talk page, as that's what it is: discussion. Just add the caveats on this page, and link to the entities (e.g. class, universal class, and so on), leaving this page as a pure and uncluttered definition of a "universe". (discuss) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. When this has been done, the template should be removed from the code. |
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.
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
- 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): $\text{I}$: 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
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets
- 1983: 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
