Set Theory

Set Theory#

Contents#

Sections#

\( \begin{aligned} & \mathbb{N}^{+} &&= \{ 1, 2, 3, \dotsc \} && \text{positive integers} \\ & \mathbb{N} &&= \{ 0, 1, 2, 3, \dotsc \} && \text{natural numbers (nonnegative integers)} \\ & \mathbb{Z}^{ -} &&= \{ -1, -2, -3, \dotsc \} && \text{negative numbers} \\ & \mathbb{Z} &&= \{ \dotsc, -3, -2, -1, 0, 1, 2, 3, \dotsc \} && \text{integers} \\ & \mathbb{E} &&= \{ \dotsc, -4, -2, 0, 2, 4, \dotsc \} && \text{even numbers} \\ & \mathbb{O} &&= \{ \dotsc, -3, -1, 1, 3, \dotsc \} && \text{odd numbers} \\ & \mathbb{P} &&= \{ 2, 3, 5, 7, 11, \dotsc \} && \text{primes} \\ & \mathbb{Q} &&= \{ p/q \mid p, q \in \mathbb{Z}, q \ne 0 \} && \text{rational numbers, expressible as the ratio of two integers} \\ & \mathbb{R} && && \text{real numbers, the continuum} \\ & \mathbb{R - Q} &&= \{ x \mid x \in \mathbb{R} \land x \not\in \mathbb{Q} \} && \text{irrational numbers, not expressible as the ratio of two integers (algebraic vs transcendental)} \\ & \mathbb{I} &&= \{ bi \mid b \in \mathbb{R} \land i = \sqrt{-1} \} && \text{imaginary numbers} \\ & \mathbb{C} &&= \{ a + bi \mid a, b \in \mathbb{R} \land i = \sqrt{-1} \} && \text{complex numbers} \\ \end {aligned} \)

\[ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \]

\( \begin{aligned} & \mathbb{Z}[x] && \text{the set of polynomials in } x \text{ with integral coefficients} \\ & \mathbb{Q}[x] && \text{the set of polynomials in } x \text{ with rational coefficients} \\ & \mathbb{R}[x] && \text{the set of polynomials in } x \text{ with real coefficients} \\ & \mathbb{Q}[x, y] && \text{the set of polynomials in } x, y \text{ with rational coefficients} \\ & \mathbb{R}[x, y] && \text{the set of polynomials in } x, y \text{ with real coefficients} \\ \end {aligned} \)

Figures#

  • [ w ] Cantor, Georg (1845-1918)

    • [ w ] (1874). On a Property of the Collection of All Real Algebraic Numbers.

  • [ w ] Dedekind, Richard (1831-1916)

  • [ w ] Russell, Bertrand (1872-1970)

  • [ w ] Zermelo, Ernst (1871-1953)

Terms#

  • [ w ] Algebra of Sets

  • [ w ] Axiom of Choice

  • [ w ] Burali-Forti Paradox

  • [ w ] Cantor’s Diagonal Argument

  • [ w ] Cantor’s Paradise

  • [ w ] Cantor’s Paradox

  • [ w ] Cardinal Number

  • [ w ] Cartesian Product

  • [ w ] Class

  • [ w ] Complement of a Set

  • [ w ] Dedekind-Infinite Set

  • [ w ] Disjoint Sets

  • [ w ] Disjoint Union

  • [ w ] Element

  • [ w ] Empty Set

  • [ w ] Equinumerosity

  • [ w ] Equivalence Class

  • [ w ] Family of Sets

  • [ w ] Finite Set

  • [ w ] Infinite Set

  • [ w ] Intersection

  • [ w ] Large Cardinal Property

  • [ w ] Multiplicity

  • [ w ] Multiset

  • [ w ] Naive Set Theory

  • [ w ] Paradoxes of Set Theory

  • [ w ] Partition of a Set

  • [ w ] Power Set

  • [ w ] Russell’s Paradox

  • [ w ] Set

  • [ w ] Set Builder Notation

  • [ w ] Set Theory

  • [ w ] Singleton

  • [ w ] Subset

  • [ w ] Symmetric Difference

  • [ w ] Transfinite Number

  • [ w ] Union

  • [ w ] Universe

  • [ w ] Zermelo-Fraenkel Set Theory

  • [ s ] Category Theory

  • [ s ] Continuity and Infinitesimals

  • [ s ] Continuum Hypothesis

  • [ s ] Large Cardinals and Determinacy

  • [ s ] Large Cardinals and Independence

  • [ s ] Set Theory

  • [ s ] Set Theory, Alternative Axiomatic

  • [ s ] Set Theory, Early Development

  • [ s ] Set Theory, Zermelo’s Axiomatization

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