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Part: 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.

Table of Contents

Motivations: 1 Explanations: 1

1. Definition: Equipotent Sets
2. Proposition: Cardinal Number
3. Definition: Finite Set, Infinite Set
4. Definition: Comparison of Cardinal Numbers
5. Chapter: Can Cardinals be Ordered?
6. Theorem: Schröder-Bernstein Theorem
7. Chapter: Simple Facts Regarding Cardinals
8. Definition: Countable Set, Uncountable Set
9. Proposition: Union of Countably Many Countable Sets
10. Proposition: Cardinals of a Set and Its Power Set
11. Proposition: Subset of a Countable Set is Countable
12. Proposition: Rational Numbers are Countable
13. Proposition: Real Numbers are Uncountable
14. Proposition: Uncountable and Countable Subsets of Natural Numbers
15. Chapter: Continuum Hypothesis

Mentioned in:

Chapters: 1
Parts: 2

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