 So the cardinal number 1 is the class of all singletons. There will be a different number 1 for each type of x. Frege, by contrast, defines the natural number 1 as the extension of a certain concept, namely being identical with the number 0, which itself is the extension of the (empty) concept of not being self identical. In axiomatic set theory the natural numbers are particular finite ordinals, in particular the series with 0 as the empty set \(\emptyset\), 1 is \(\{ 0 \}\), 2 is \(\{0, 1 \}\), and so on. This construction is named the von Neumann ordinals.