We will now introduce cardinality.
Cardinality is a way of measuring the ‘size’ of sets by the number of elements in them. Before we have a precise notion of cardinality, we define a relation
A set A is equinumerous to a set B if there exists a bijection f\colon A \to B.
For any sets A,B and C:
(i) A \approx A.
(ii) If A \approx B, then B \approx A.
(iii) If A \approx B and BB \approx C then A \approx C.
Thus, theres sorta an equivalancy relation to equinumeroisty.
No set is equal in cardinality to it’s powerset.
A set is finite if it is equinumerous to some natural number.
Cardinal numbers will be some element of the set \text{card } A for some set A. We define \text{card } \omega = \aleph_0.
The set of sets with cardinality k is too large to be a set.
Any subset of a finite set is finite.
Let \kappa and \lambda be cardinal numbers.
(a) \kappa + \lambda = card (K \cup L) where K and L are any disjoint sets of cardinality \kappa and \lambda.
(b) \kappa \cdot \lambda = card (K \times L) where K and L are any sets of cardinality \kappa and \lambda.
(c) \kappa^\lambda = card (K^L) where K and L are any disjoint sets of cardinality \kappa and \lambda.
If C is a proper subset of a natural number n, then C \approx m for some m less than n.
Any subset of a finite set is finite.