Definition:Universe (Set Theory)/Zermelo-Fraenkel Theory

From ProofWiki
Jump to navigation Jump to search


It has been suggested that this page or section be merged into Definition:Universal Class/Zermelo-Fraenkel Theory.
In particular: Address subtle differences
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Mergeto}} from the code.



This article, or a section of it, needs explaining.
In particular: 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.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


Universal Set in Zermelo-Fraenkel Set Theory

If the universal class is allowed to be a set $\mathbb U$ in ZF(C) set theory, then a contradiction results.

One equivalent of the axiom of specification states that:

$\forall z: \forall A: \paren {A \subseteq z \implies A \in \mathbb U}$


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.



This page needs proofreading.
Please check it for mathematical errors.
If you believe there are none, please remove {{Proofread}} from the code.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Proofread}} from the code.



Work In Progress
In particular: 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".
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{WIP}} from the code.



Sources


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
In particular: Determine whether this discussion appears in these works
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Universe_(Set_Theory)/Zermelo-Fraenkel_Theory&oldid=624393"