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**ThePerfectHacker** This is a little bit off topic but you might it interesting we define $\displaystyle 2$ to be $\displaystyle \{ 0,1\}$ where $\displaystyle 0=\emptyset$ and $\displaystyle 1=\{0\}$. This is what Plato said. Thus, there is no need to write $\displaystyle \bar 2 = \{ 0,1\}$ because in fact $\displaystyle 2=\{0,1\}$.

This might look strange but this is how we define natural numbers. We define $\displaystyle 0 = \emptyset$. We define $\displaystyle 1=\{ 0\} = \{ \emptyset \}$. We define $\displaystyle 2=\{0,1\} = \{ \emptyset, \{ \emptyset\} \}$. We define $\displaystyle 3=\{0,1,2\}=\{ \emptyset, \{ \emptyset\}, \{\emptyset, \{\emptyset\} \} \}$. We would like to say natural numbers are repetitions of this form. However, this is not a formal enough statement. So this is what we do. Let $\displaystyle x$ be a set and define $\displaystyle x+1 = x\cup \{ x\}$. Thus is follows $\displaystyle 0 = \emptyset, 1 = 0+1,2=1+1,3=2+1$. We cannot say "and so on" because that is not a formal term. To do this we define an inductive set to be a set $\displaystyle I$ such that $\displaystyle 0\in I$ and if $\displaystyle x\in I$ then $\displaystyle x+1\in I$. One of the axioms we place into Set Theory is called Axiom of Infinity which says "an inductive set exists", the reason we do this is because using the simpler axioms we cannot construct inductive sets and we would like to make set theory more interesting by allowing infinite sets. Finally, since there is an inductive set $\displaystyle I$ define $\displaystyle \mathbb{N} = \{ x\in I | x \mbox{ is in every inductive set } \}$. This set $\displaystyle \mathbb{N}$ are called the natural numbers and this is a purely formal construction of the naturals.