TITLE: How can we define the set $\{\{\}, \{\{\}\}, \{\{\{\}\}\}, ...\}$?
QUESTION [6 upvotes]: How do you define the set of elements which can be made by repeatedly wrapping the null set in another set: $\{\{\}, \{\{\}\}, \{\{\{\}\}\}, ...\}$?
I've tried $A = \{x\ |\ x=\emptyset \lor \exists y\in A : x = \{y\}\}$, but that would give us a recursive formula, because $y\in A$ is replaced by that same formula.
Is this set definable, and if yes, how?

REPLY [4 votes]: How about:
$$\{x\mid\forall A,(\{\}\in A\land(\forall a\in A,\\\{a\}\in A))\Rightarrow x\in A\}$$
To make that easier to read, let me define the following. A set $A$ is upwards if $a\in A$ implies $\{a\}\in A$. (I made up this word just now.) A set $A$ is called totally upwards if it's upwards and it contains $\{\}$. (I also just made this up.) Then the above reads:
$$\{x\mid\forall A,\\A\text{ is totally upwards }\Rightarrow x\in A\}$$