Page 53 - Textos de Matemática Vol. 38

ORDINALS, COMPUTATIONS, AND MODELS OF SET THEORY 45
2 Ordinal numbers
Set theory, naively or axiomatically, is a natural, strong and convenient theory which can be used as a foundation for all of mathematics. I.e., the stan- dard notions can be defined naturally and their usual properties can be shown naively or from the axioms.
Numbers, in particular natural and real numbers, are the most impor- tant mathematical notions. The real numbers can be obtained from the natural numbers in the usual set theoretic way: natural numbers → rational number → Dedekind cuts ≡ real numbers. In set theory one considers an infinitary exten- sion of the natural numbers: Cantor’s ordinal numbers allow to count beyond the natural numbers into the transfinite. We shall use a formalization which is usually associated with John von Neumann. We motivate the formalization by introducing (some) natural numbers in a seemingly ad hoc way.
In a sense which can be made precise, sets can iteratively be generated from the empty set ∅. We also base numbers on the empty set. Define recursively
0=∅
1 = {0}
2 = {0,1}
3 = {0,1,2}
.
n+1 = {0,1,...,n} .
(2.1)
Obviously, the n-th set n has exactly n elements and we have chosen adequate representatives for the intuitively given “standard” numbers. We state some facts about our numbers which will lead to a general definition of number.
Proposition 2.1. Let m, n be numbers as above. Then m ∈ n iff the corresponding standard numbers satisfy m < n. So for the above numbers, ∈ is isomorphic to the standard ordering < on natural numbers.
2.1 Definitions
Definition 2.2. A set or class A is transitive, Trans(A), iff ∀u,v(u∈v∧v∈A →
u ∈ A). Obviously:
Proposition 2.3. Let m be a number as in (2.1). Then a) m is transitive,
b) every element of m is transitive.