Natural numbers from the empty set
Start with nothing. Build each number from the ones that came before it.
Build a number
The von Neumann construction4 = {0, 1, 2, 3}
Each inner circle is one member of the set. Select a member below to explore its construction.
Look inside
Every number here is a set. Open the braces to see what it contains, all the way down to the empty set.
The rule
0 = ∅
n + 1 = n ∪ {n}
The next number contains the previous number and all of its members.Why the empty set matters
Zero has no members, so it is the empty set: ∅. One contains zero, so 1 = {∅}. These are different sets: one has no elements, and the other has one.
Each new number contains every smaller natural number. Nothing new needs to be invented beyond sets and membership.