 Von Neumann (1923) defined a canonical representation of ordinal numbers, using the fact that the ordinals are themselves well-ordered. Each ordinal is represented by the set consisting of all smaller ordinals. Thus 0 is represented by the empty set \(\emptyset\), 1 by the set containing the empty set \(\{\emptyset\}\), 2 by the set containing both of those \(\{\emptyset,\{\emptyset\}\}\), and so on. \(\omega\) is then the set containing all of these finite ordinals \(\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots\}\), \(\omega+1\) is the set containing it as well \(\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots,\omega\}\), and so on.