First infinite von Neumann ordinal
Set
Idea
This is probably the most straightforward way to set up a countably infinite set.
Elaboration
The first requirement says that all elements of ωN are either ∅ or a successor of another set. The second guaranties that there are no superfluous sets in ωN, apart from the ones which are e.g. required to making ωN an ordinal. Put together the axiom says that ωN contains ∅ and for each established element m, it also inductively contains all successors succ m≡m∪{m}.
Notation
To model the natural numbers, one can make the following identifications:
- 0≡∅
- 1≡succ (0)=∅∪{∅}={}∪{0}={0}
- 2≡succ (1)=1∪{1}={0}∪{1}={0,1}
- 3≡succ (2)=2∪{2}={0,1}∪{2}={0,1,2}
- 4≡succ (3)=…
Using our common language conception of natural numbers we can say: Each number contains the numbers which are less than itself, i.e. n is {0,1,2,3,4,…,n−1}. Being an ordinal, we can model the order relation of the natural numbers via set inclusion k<n≡k∈n. This also gives us the familiar ordinal order.