Discusses the different types of numbers used in set theory: ordinal and cardinal numbers. Includes concepts like transfinite induction and arithmetic operations on transfinite numbers.
Sets: A collection of objects or elements.
Elements: The objects that form a set.
Power Set: The collection of all subsets of a set.
Cardinality: The number of elements in a set.
Finite Set: A set with a countable number of elements.
Infinite Set: A set with an uncountable number of elements.
Ordinal Numbers: A number that represents the order or position of an element in a set.
Successor function: A function that takes an element of a set and returns the next element.
Transfinite induction: A method used to prove properties of infinite sets.
Well-ordering principle: Every nonempty set of ordinal numbers has a least element.
Cardinal Numbers: A number that represents the size or quantity of a set.
Cardinality of Infinite Sets: The concept of comparing the sizes of infinite sets.
Cantor's Diagonal Argument: A proof that there are different sizes of infinite sets.
Axiom of Choice: A controversial axiom used in set theory that allows the selection of an element from each set in a collection.
Continuum Hypothesis: A statement that there is no set whose cardinality is strictly between that of the integers and the real numbers.
Zermelo-Fraenkel Set Theory: A set of axioms that form the foundation of modern set theory.