How to show $\{\{x\}\} \neq x$?

Question

Let $x$ be any set. Using ZFC axioms, how to show $(x,x)=\{\{x\}\} \neq x$?

Similar question: $\bigcup x \neq x?$

Solved questions: $x \neq P(x)$ ($x$ is a subset of $x$ but $x \notin x$ by the Foundation axiom)

$\{x\}=\{x,x\}\neq x$ (by the Foundation axiom)

score 15 · Answer 1 · answered Apr 19 '13 at 04:53

15

It seems the following:

If $\{\{x\}\}=x$ then put $y=\{x\}$. Then $y\ni x\ni y\dots $, that is impossible.

$\bigcup\varnothing=\varnothing$.

answered Apr 19 '13 at 04:53

Alex Ravsky

106,166

user642796 · Accepted Answer · 2013-04-19T06:45:23.870

14

If $x = \{ \{ x \} \}$ then $x \in \{ x \} \in \{ \{ x \} \} = x$, and so the set $\{ x , \{ x \} \}$ has no $\in$-minimal element (contradicting Foundation).

The second statement is not true. It is clearly not true for $\varnothing$, and as Andres Caicedo has mentioned below, it is not true for plenty of nonempty sets.

edited Apr 19 '13 at 06:45

answered Apr 19 '13 at 04:53

user642796

53,641

5

$\bigcup x=x$ is satisfied by plenty of sets. For example, $x=\omega$, or $x=V_{\omega_3+\omega_1\cdot 2}$. – Andrés E. Caicedo Apr 19 '13 at 05:21
2

@Andres: Crap crap crap crap. (And thank-you very much for pointing out what was clearly obvious to many, but not to myself.) – user642796 Apr 19 '13 at 06:43

How to show $\{\{x\}\} \neq x$?

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