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The definition of the natural numbers I like is the set $\{\emptyset,\{\emptyset\},\{\{\emptyset\}\},...\}$ where $\emptyset=0$ and $n+1=\{n\}$. The simplest definition of the real numbers is the power set of the natural numbers, though I was never told how each number is defined. Are there definitions of the integers and rational numbers where $N\subset Z\subset Q\subset R$ holds? All of the other definitions I've seen just embed $N$ in $Z$, but I would like $N$ to be a subset of $Z$.

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