Cardinal Numbers 

In this part, we will be dealing with cardinal nubers or short cardinals. Cardinals are a mathematical clarification of the intuitive concept of how many elements a set has. For finite sets, it is possible to “count” the elements and tell the number corresponding to the number of elements in this set, or its cardinality. It turns out that cardinals represent the cardinalities of both, finite and infinite sets. But not only this. Cardinals do not only represent the cardinalities of other sets of but they are sets themselves. In the following text, we will see, how this all fits together.

| | | | created: 2014-02-20 23:34:09 | modified: 2020-06-07 16:31:29 | by: bookofproofs

1.Definition: Equipotent Sets 

2.Proposition: Cardinal Number 

3.Definition: Finite Set, Infinite Set 

4.Definition: Comparison of Cardinal Numbers 

5.Can Cardinals be Ordered? 

6.Theorem: Schröder-Bernstein Theorem 

7.Simple Facts Regarding Cardinals 

8.Motivation: Cantor's Astonishing Discoveries Regarding the Cardinals of Infinite Sets 

9.Explanation: Transitive Set and Countability - Natural Numbers Have the Smallest Infinite Cardinality 

10.Definition: Countable Set, Uncountable Set 

11.Proposition: Union of Countably Many Countable Sets 

12.Proposition: Cardinals of a Set and Its Power Set 

13.Proposition: Subset of a Countable Set is Countable 

14.Proposition: Rational Numbers are Countable 

15.Proposition: Real Numbers are Uncountable 

16.Proposition: Uncountable and Countable Subsets of Natural Numbers 

17.Continuum Hypothesis