Infinite Sets and Set Theory

Infinitism is the philosophical school of thought that asserts the existence of the actual infinite, such as an infinite number of mathematical objects or an infinite universe.

Leopold Kronecker was a German mathematician who rejected the concept of the actual infinite and argued that only finite mathematical objects exist.

Cantor's Theorem, also known as the Cantor-Bernstein-Shroeder Theorem, states that for any two sets, A and B, either A is equinumerous to a subset of B, or B is equinumerous to a subset of A, or there exists a set C that is equinumerous to A and a proper subset of B.

Russell's Paradox, also known as the Russell-Zermelo Paradox, arises from the consideration of the set of all sets that do not contain themselves. If such a set exists, it leads to a contradiction, as it must contain itself if and only if it does not contain itself.

The Bolzano-Weierstrass Theorem, also known as the Intermediate Value Theorem for continuous functions, states that if a function is continuous on a closed interval, then for any value between the function values at the endpoints of the interval, there exists a point in the interval where the function takes that value.

An uncountable set is a set that cannot be put into one-to-one correspondence with the set of natural numbers. The most famous example of an uncountable set is the set of real numbers.

Cantor's Theorem, also known as the Cantor-Bernstein-Shroeder Theorem, states that for any two sets, A and B, either A is equinumerous to a subset of B, or B is equinumerous to a subset of A, or there exists a set C that is equinumerous to A and a proper subset of B.

A countable set is a set that can be put into one-to-one correspondence with the set of natural numbers. Examples of countable sets include the set of integers, the set of rational numbers, and the set of algebraic numbers.

The Completeness Axiom, also known as the Supremum Axiom, states that every non-empty set of real numbers that is bounded above has a least upper bound. This axiom is essential for the development of real analysis.

An uncountable set is a set that cannot be put into one-to-one correspondence with the set of natural numbers. The most famous example of an uncountable set is the set of real numbers.

The Completeness Axiom, also known as the Supremum Axiom, states that every non-empty set of real numbers that is bounded above has a least upper bound. This axiom is essential for the development of real analysis.

A countable set is a set that can be put into one-to-one correspondence with the set of natural numbers. Examples of countable sets include the set of integers, the set of rational numbers, and the set of algebraic numbers.

The Completeness Axiom, also known as the Supremum Axiom, states that every non-empty set of real numbers that is bounded above has a least upper bound. This axiom is essential for the development of real analysis.

An uncountable set is a set that cannot be put into one-to-one correspondence with the set of natural numbers. The most famous example of an uncountable set is the set of real numbers.

The Completeness Axiom, also known as the Supremum Axiom, states that every non-empty set of real numbers that is bounded above has a least upper bound. This axiom is essential for the development of real analysis.