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I have some interest in alternative formulations of set theory which blatantly admit {x}={{x}}. What gave rise to such interest is the following. Take the set equation A={A}. The RHS dictate that {A} contains one element. So since A={A}, A is also of cardinality 1. We therefore get {x}={{x}} which has no solution. But if we admitted that {x}={{x}}, we could have a solution: A can be any set of a single element. Under such formalism, therefore, set equations become more "algebrically closed".

I'll be thankful if anyone can point me to such existing formalisms of set theory as I didn't find any.

Edit: we will need to give up the regularity axiom https://en.wikipedia.org/wiki/Axiom_of_regularity#No_set_is_an_element_of_itself but that's the edge of my knowledge

Andrés E. Caicedo
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Troy McClure
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    Are you looking for set theories where $\forall x.{x}={{x}}$ holds, or just ones where there are particular $x$ that satisfy ${x}={{x}}$? – hmakholm left over Monica Jun 21 '18 at 10:47
  • as long as we can have a solution for A={A}, I wouldn't mind really. But consider the following argument: physically, {x}={{x}} indeed. The set that contains the sun is no more than the sun. So one could philosophically consider it for any x. – Troy McClure Jun 21 '18 at 10:49
  • If $A={A} \forall A$, then $\emptyset ={\emptyset}$, so $0=1$. So you could not use the Neumann definition of natural numbers. – Botond Jun 21 '18 at 10:54
  • @TroyMcClure, based on this comment it seems that you are interested in something like second-order logic rather than a set theory: this allows one to talk about objects and collections of objects, but nothing else/deeper. A set theory where the equation ${a} = {{a}}$ holds more generally based on the idea that it "ultimately" contains the same elements would be very weird; in particular, from ${a} = {{a}}$ you should be able to conclude $a = {a}$, which is false when $a$ is the empty set. – Mees de Vries Jun 21 '18 at 10:54

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There are well-studied set theories where the equation $x=\{x\}$ has a solution (and therefore also $\{x\}=\{\{x\}\}$, of course). Such a solution is sometimes called a Quine atom.

Look for non-well-founded set theory, and for the anti-foundation axiom in particular.

On the other hand, having $\forall x.\{x\}=\{\{x\}\}$ would require major changes to our conception of sets, and I'm not sure the result would be recognizable as set theory at all.

In particular, the notation $\{x\}$ is usually taken to mean "a set whose only element is $x$", so if $\{x\}=\{\{x\}\}$, then we have one set whose only element is simultaneously $x$ and $\{x\}$. This means that $x=\{x\}$, or again that the only element of $x$ is $x$ itself.

So in such a theory, if you want any set that contains two different things (or just something different from itself), you would need to give up the usual interpretation of $\{\cdots\}$ in terms of $\in$.

Trying to work out what you want to put instead, you might end up reinventing some form of mereology.

hmakholm left over Monica
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  • And also ${\varnothing}={{\varnothing}}$ would require a new way of thinking about $\varnothing$ and about $=$. – Asaf Karagila Jun 22 '18 at 11:40
  • @AsafKaragila: I suspect after redefining ${\cdots}$ such that $\forall x. {x}={{x}}$ is possible, we would end up having ${\varnothing}$ mean the same thing as $\varnothing$ anyway. (Lots of confused freshmen rejoice?) – hmakholm left over Monica Jun 22 '18 at 12:28