PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Logic and Foundations → Set Theory ↓


Ordinals and Cardinals


In the foundations of mathematics, particularly in set theory, understanding ordinals and cardinals is important for understanding the nature of infinite sets and the hierarchy of different sizes of infinity. These two concepts play a central role in how we understand the order, size, and basic structure of mathematical universes.

Introduction to set theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It provides a basic framework for dealing with objects that can be collected into groups or sets, and it is the basis for much of modern mathematics. Set theory helps explain what a collection of mathematical entities is and explains how these collections relate to one another.

Understanding cardinals

Cardinals are related to the size of a set. More precisely, the cardinality of a set measures the number of elements in that set. For finite sets, it is simply a count of elements. For infinite sets, the concept of cardinality becomes much richer and requires more sophisticated tools to understand it.

To illustrate cardinality consider two sets:

    A = {1, 2, 3}
    B = {x, y, z}

Both sets A and B have cardinality 3 because each of them has three elements.

Finite and infinite cardinals

The concept of counting extends from finite sets to infinite sets, although in a more subtle way. For finite sets, counting is straightforward, but with infinite sets, we enter a realm of mathematics that is beyond intuitive calculation. The smallest infinite cardinal is denoted as ℵ0 (aleph-null), which is the cardinality of the set ℕ of natural numbers.

Infinite cardinals allow us to compare the size of different infinite sets. For example, the cardinality of the set ℝ of real numbers is larger than the set ℕ of natural numbers, even though both are infinite. This is a fascinating aspect of set theory and leads us to investigate the concept of bijection to compare infinite sets.

Understanding ordinals

Ordinals extend the concept of ordering to sets. They not only serve as a means of distinguishing between different kinds of infinities, but also provide a hierarchy. Ordinals are essential for describing the type of order exhibited by sets, especially those that are well-ordered.

Well-ordered sets

A set is well-ordered if every non-empty subset has a minimal element under its order. This property is important in the study of ordinal numbers because every ordinal set can be associated with a unique ordinal number. For example, the natural numbers are well-ordered by the general "less than" relation.

Construction of ordinals

Ordinals are constructed using transfinite induction, and they start with the smallest ordinal, 0, and proceed with the successor operation. Let's construct some ordinals:

    0, 1, 2, 3, ..., n, n+1, ...

These are finite ordinal sequences. For infinite sequences, we use:

    ω, ω+1, ω+2, ...

where ω denotes the first transfinite ordinal corresponding to the order type of the natural numbers.

Visual representation of ordinal numbers

Understanding ordinal numbers can be enhanced through visualization. Consider representing ordinal numbers as a sequence on a line:

,ω

Here, the sequence denotes finite ordinals, ending at ω, the first limit ordinal.

Cardinal arithmetic

Cardinal arithmetic deals with the operations of addition, multiplication, and exponentiation on cardinals. Unlike ordinary arithmetic, cardinal arithmetic behaves differently due to the nature of infinite sets.

Addition and multiplication

For finite sets, addition and multiplication work as expected. For example, if |A| = 3 and |B| = 5, then:

    |A ∪ B| = |A| + |B| = 8

Multiplication for finite sets is similarly simple:

    |A × B| = |A| × |B| = 15

Infinite cardinal arithmetic

For infinite sets, the operations can be surprising. Consider two sets, both of which have cardinality ℵ0:

    ℵ0 + ℵ0 = ℵ0
    ℵ0 × ℵ0 = ℵ0

These results show that adding or multiplying two countably infinite sets still gives a countably infinite set.

Visual representation of cardinals

Understanding infinity

If we think of infinite sets as collections that cannot be exhaustively listed or finitely numbered, it is helpful to think of these sets as collections that extend endlessly in all directions.

,

Relation between ordinals and cardinals

Ordinals and cardinals are connected in such a way that ordinal numbers can represent the order type of a well-ordered set, while cardinal numbers represent the size of the set. A cardinal number can be seen as an elementary ordinal, denoting the smallest ordinal of that particular size.

Through this relationship, each cardinal is associated with a unique ordinal number. The beautifully structured world of ordinal and cardinal numbers helped mathematicians like Cantor, who revolutionized the understanding of infinity, bring precision and insight to the study of infinite sets.

Closing thoughts

When diving into the depths of set theory, ordinals and cardinals provide profound insights into mathematics and the nature of infinity. Understanding these concepts clarifies the hierarchy and scope of mathematical universes and sets an exciting and comprehensive path for exploring infinite structures.


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