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
- Definition: Equipotent Sets
- Proposition: Cardinal Number
- Definition: Finite Set, Infinite Set
- Definition: Comparison of Cardinal Numbers
- Chapter: Can Cardinals be Ordered?
- Theorem: Schröder-Bernstein Theorem
- Chapter: Simple Facts Regarding Cardinals
- Definition: Countable Set, Uncountable Set
- Proposition: Union of Countably Many Countable Sets
- Proposition: Cardinals of a Set and Its Power Set
- Proposition: Subset of a Countable Set is Countable
- Proposition: Rational Numbers are Countable
- Proposition: Real Numbers are Uncountable
- Proposition: Uncountable and Countable Subsets of Natural Numbers
- Chapter: Continuum Hypothesis
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