11.4 Von Neumann’s model for \(\mathbb N\)

In about 1922, John von Neumann proposed the following model, \((N,t,z)\), for \(\mathbb N\). \(N\) is a set whose elements are themselves sets. We let \(z=\emptyset\) and for \(X\in N, t(X) = X\cup\{X\}\). We then see that \[ \begin{array}{rcl} t(\emptyset)& = &\emptyset\cup\{\emptyset\} = \{\emptyset\}\\ t(t(\emptyset))& = &\{\emptyset\}\cup\{\{\emptyset\}\}=\{\emptyset,\{\emptyset\}\}\\ t(t(t(\emptyset)))& = &\{\emptyset,\{\emptyset\}\}\cup\{\{\emptyset,\{\emptyset\}\}\} = \{\emptyset,\{\emptyset\},\{\{\emptyset,\{\emptyset\}\}\}\} \end{array} \] and so on. It is possible to show that this model satisfies Peano’s axioms. So \[ N=\{\emptyset,\{\emptyset\},\{\{\emptyset,\{\emptyset\}\}\},\ldots\}. \] If, as normal, we denote \(\emptyset\) by \(0\), \(t(\emptyset)\) by \(1\) and so on then we see that \[ 1=\{0\}, 2=\{0,1\}, 3=\{0,1,2\}\text{ etc} \] and in general \(n=\{0,1,\ldots,n-1\}\). This model plays an important role in the theory of (infinite) ordinal numbers.