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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

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)