TITLE: What interpretations exists for the singleton of the singleton of the empty set $\{\{\varnothing\}\}$ and its singleton $\{\{\{\varnothing\}\}\}$?
QUESTION [5 upvotes]: What I know

I know from Peano's axioms that the empty set is equivalent to the natural number $0$ and that the singleton of the empty set is equivalent to the natural number $1$. ( http://www.proofwiki.org/wiki/Peano%27s_Axioms_Uniquely_Define_Natural_Numbers , http://www.proofwiki.org/wiki/Axiom:Peano%27s_Axioms)
I know from Kuratowski that the set $\{\{A\}\}$ is equivalent to the ordered pair $\langle A,A\rangle$.

What I want to know

Are there any alternative interpretations of $\{\{A\}\}$ other than Kuratowski's?
How can we express an ordered homogeneous triple (e.g. Would $\langle A,A,A\rangle$ be equivalent to $\{\{\{A\}\}\}$?)
It seems like the set $\{\{\varnothing\}\}$ can be interpreted as either $\{1\}$ or $\langle\varnothing,\varnothing\rangle$. Should $\{\{\{\varnothing\}\}\}$ be interpreted as $\langle\varnothing,\varnothing,\varnothing\rangle$ or as $\langle 1,1\rangle$.  Is it possible that these tuples are in some way equivalent?

Note, re: Peano's Axioms
My understanding of Peano's axioms is consistent with the formulation at WolframAlpha, which is expressed as the 5 postulates:

$0$ is a number.
If $a$ is a number, the successor of $a$ is a number.
$0$ is not the successor of a number
Two numbers of which the successors are equal are themselves equal.
If a set $S$ of numbers contains zero and also the successor of every number in $S$, then every number is in $S$.

These axioms are expressed in set notation at http://www.proofwiki.org/wiki/Axiom:Peano%27s_Axioms / http://www.proofwiki.org/wiki/Peano%27s_Axioms_Uniquely_Define_Natural_Numbers .

REPLY [10 votes]: You're taking the Kuratowski ordered pair definition too seriously. It doesn't have any intrinsic intuitive meaning, and in particular does not tell us anything deep about the sets it chooses to represent ordered pair. Its only role is as a technical detail in the proof of

ORDERED-PAIR METATHEOREM. There exists a formula $\phi(x,y,p)$ with three free variables such that

For all $x$ and $y$ there is exactly one $p$ such that $\phi(x,y,p)$.
If $\phi(x,y,p)$ and $\phi(x',y',p)$, then $x=x'$ and $y=y'$.

are theorems of set theory.

In the entire rest of the development of set theory (including the very formulation of several of the axioms), and of the development of the rest of mathematics as an application of set theory, we forget completely which formula this $\phi(x,y,p)$ actually is and just write it as $\langle x,y\rangle=p$ instead.
This involves taking care that we never depend on any properties of ordered pairs other than what the metatheorem guarantees -- in particular we never put ordered pairs and other things into the same set and expect to be able to tell them apart later, because a priori anything might be a pair. We could have proved the ordered-pair metatheorem for a different $\phi$ -- for example one that represents the particular pair $\langle \omega_3,\{42,19\}\rangle$ as $\varnothing$ and everything else as Kuratowski pairs in reverse order -- and that should not make any difference at all for the rest of the development.
So yes, it is true in a technical sense that $\langle 0,0\rangle
$ and $\{1\}$ are the same set. But that is just a coincidence -- an implementation detail -- and we deliberately don't depend on such details when we do mathematics.
It makes very good sense, as Asaf suggests, to view this as a type system. In that view, asking whether $A=B$ when we know $A$ as an ordered pair, and $B$ as a set with particular members, is a type error which produces an implementation-dependent (even if definite) truth value.