Are there "interesting" (that is non-trivial, for example not containing only one set) set theories with one element set being equal to their element ($\{x\}=x$ for every $x$)?

This question arose from the practical problem: Is it possible without "troubles" (such as contradictions) to consider an RDF term denoting a transformation between XML namespaces as an one-element set containing this term? If we can consider them equal, it makes shorter the notation, as we do not need to define an one-element set in this case but use the transformation term itself to denote this set. To define one term less in this case.

  • I'm not aware of any theory where every set $x$ is equal to $\{x\}$ (for one, this would make the notion of cardinality pointless) but throughout mathematics people will abbreviate a singleton $\{x\}$ with just $x$, though they are always implicitly meaning the singleton $\{x\}$, treating $x$ almost like a urelement. – Hayden Aug 11 '17 at 0:10
up vote 8 down vote accepted

Yes. If you take out one axiom (the Axiom of Regularity, which roughly disallows sets to be nested inside themselves), then ZFC is perfectly happy with the existence of sets $x$ such that $x = \{x\}$. Such sets $x$ are usually known as Quine atoms.

In fact there are many well-known set theories that explicitly allow the existence of Quine atoms, sometimes as a matter of principle --- New Foundations, for instance.

Of course it will never be the case for all $x$ that $x = \{x\}$. This would give a contradiction. Specifically, we can prove that $\varnothing \ne \{\varnothing\}$. If you wanted to change this, you would have to change the very definition of membership or equality of sets.

Given your intended computer programming application, I will also mention that there are programming languages which treat $x$ and $\{x\}$ as the same thing. This is not a problem, there is no contradiction, because "sets" in these languages are much more restricted than the sets of set theory. Typically, $\{\varnothing\}$ will not be an allowed set, and in general there is only a single level of nesting (i.e., no sets-of-sets).

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    Beat me to it. +1. Also, New Foundations guarentees the existence of more than 1 set theories. (To answer the non-triviality). – Dair Aug 11 '17 at 0:13
  • @Dair I assume you mean more than 1 Quine atom. How are the Quine atoms constructed? I have thought about it only briefly. – 6005 Aug 11 '17 at 0:33
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    In this vein, there's a whole book on this, or Peter Aczel's seminal monograph which led to a lot of research into the categorical notion of a coalgebra. – Derek Elkins Aug 11 '17 at 1:07
  • What programming languages are you talking about? What you say doesn't apply to any programming language that I am familiar with. – Rob Arthan Aug 12 '17 at 23:51
  • @RobArthan Alloy developed at MIT is the example that I am familiar with and that justifies my comment. However, I believe it happens in many database programming languages, in particular any that are based on Relational algebra and employ relational join. Disclaimer that it is not exactly my area of expertise. Thanks for your question. – 6005 Aug 13 '17 at 14:07

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