8

As far as I'm aware, when doing mathematics (apart from when doing axiomatic set theory) the assumption that all objects are sets, and that there are no urelements, is never actually used. When reasoning about a group, or a topological space for instance, it never matters that the elements of the structure are themselves sets rather than urelements. Are there any general results to this effect - showing that if one works in ZA say (Zermelo set theory in which atoms/urelements may exist), one can establish the same mathematical results in some sense as if one works in Z (Zermelo set theory)? For instance if $T$ is a second order theory and $\phi$ a statement in the language of $T$ such that $\text{Z}\vdash(T\vDash \phi)$, does it follow that $\text{ZA}\vdash(T\vDash\phi)$?

Blunka
  • 925
  • "urelements" ?... – Jean Marie Mar 05 '20 at 18:58
  • 1
    Objects which aren't sets, and don't have members. They can be distinct despite each of them having no members (so don't satisfy extensionality) https://en.m.wikipedia.org/wiki/Urelement – Blunka Mar 05 '20 at 19:24
  • Thnk you for your answer. – Jean Marie Mar 05 '20 at 20:17
  • Interestingly, there are technical situations where urelements are actually desirable - this is an important theme in Barwise's book Admissible sets and structures. Roughly speaking, given any structure $M$ we can build a model of an appropriate set theory "over" it, with urelements corresponding to the elements of $M$. Using the set theory KPU here leads to useful applications in (higher) computability theory. – Noah Schweber Mar 05 '20 at 23:10
  • 1
    As to your question, this doesn't answer it exactly but the Jech-Sochor embedding theorem (see e.g. the discussion here) implies that the answer is yes for ZF - Z, however, is vastly weaker, and while I suspect the result is the same I suspect the argument needs to be much more careful. Since you ask specifically about Z, though, this doesn't answer your question. (Although that said, why do you care about Z as opposed to ZF? The latter is a lot more natural in my opinion, and certainly better understood and more used ...) – Noah Schweber Mar 05 '20 at 23:14
  • 1
    Note that both Asaf's answer and my comments are emphasizing that sometimes urelements are more useful than their absence! I think in light of that it's worth saying a bit about why ZFA didn't catch on. In my opinion, the reason is twofold: $(i)$ it's not until you start doing computability theory that they're useful as more than an intuitive device, so it's not really compelling, just neat; and $(ii)$ unlike with pure sets there's no sense that there's a "right" collection of urelements. E.g. should the urelements form a set or class? Should they be linearly orderable? Etc. – Noah Schweber Mar 06 '20 at 16:28
  • 1
    Incidentally, let me know if you'd like me to expand on my comments in an answer. I don't think they're exactly addressing your question, but if you'd like me to make them more formal I'm happy to do so. – Noah Schweber Mar 06 '20 at 16:29
  • @Noah: I'd be happy to upvote your answer should you write one (which you should). – Asaf Karagila Mar 06 '20 at 21:29

1 Answers1

2

To some extent, but not quite there, you're right. Urelements (or "atoms") are kinda irrelevant. When you want to reason about structure, the underlying set is irrelevant, and so whether or not your set is pure or with urelements is not important.

Nevertheless, if you want to reason about the universe, urelements may play a role. In the study of the Axiom of Choice, for example, using urelements is an easy approach which is considered clearer than applying forcing-style arguments. And we even have abstract transfer theorems letting us move from urelements to results in ZF. But it is not true that everything can be transferred.

For example, "The power set of every ordinal can be well-ordered" implies the Axiom of Choice in ZF, but not in ZFA. If you want to learn more on these kind of statements, Jech's "The Axiom of Choice" has a nice chapter about this which covers the basics.

Other examples of this flavour come from a proper class of urelements (e.g. here and there on this very website) being used to separate other forms of Global Choice or Replacement-like axioms. (One thing to note is that we can replace Urelements by Quine atoms, i.e. sets satisfying the equation $x=\{x\}$, which is why we can think of pure sets as something satisfying the Axiom of Regularity/Foundation.)

Asaf Karagila
  • 393,674