TMU = The Math Universe Remix and add ONE CIRCLE to the tree. Connect it with things that are closely related. ========== The first axiom in Peano arithmetic proposes that "0 is a natural number", indicating the property of natural numbers, so node #000 has been connected to node #001, the set of natural numbers. In first order set theory, the number 0 is equivalent to the empty set (∅). The successor function is formally defined as S(n) := n ∪ {n}, which means that, given n, yield a set that contains the contents of n and n itself. Define ∅ as 0 S(0) = 0 ∪ {0} = {} ∪ {{}} = {{}} = {0} ({0} is defined as 1. There's nothing in the set {}, so nothing gets added in {{}}, and it equals {0}) S(1) = 1 ∪ {1} = {0, 1} (define it as 2) S(2) = 2 ∪ {2} = {0, 1, 2} (and so on) All of these numeric definitions are also defined with the natural number property in set theory, which means that the set of natural numbers is formally defined by the repeated use of this successor function starting at 0, creating positive integers such as 2, 6, 18, etc.
