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.

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

Chapters: 1
Parts: 2

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