Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Mathematical Logic and Foundations → Set Theory ↓


Axiomatic Systems


In the vast field of mathematics, one of the fundamental approaches to understanding mathematical concepts is through axiomatic systems. An axiomatic system is a set of axioms or basic principles from which other truths can be derived. Axioms are fundamental statements or statements that are considered true without proof. The main purpose of using axiomatic systems is to provide a framework where the derivation of each proof can be traced back to these primary principles.

Understanding axiomatic systems

An axiomatic system typically includes the following:

  • Axioms: Basic, self-evident truths from which theorems can be proved.
  • Undefined terms: Terms that are used in axioms and theorems but are not defined within the system.
  • Definitions: New terms that are defined using axioms and already defined terms.
  • Theorem: Propositions proven based on axioms and previously proven theorems.
  • Logical rules: Rules of inference that are used to deduce theorems from axioms.

When analyzing an axiomatic system, the syllogism and validity of each statement can be tested, ensuring that the mathematics built within this system stands on solid ground.

Major examples of axiomatic systems

Euclidean geometry

One of the earliest known uses of an axiomatic system is Euclidean geometry, which is based on five principles that Euclid introduced in his work "Elements."

  1. A straight line can be drawn by joining any two points.
  2. A straight line segment can be extended indefinitely into a straight line.
  3. Given any straight line segment, a circle can be drawn with the segment as radius and one of its endpoints as centre.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third line such that the sum of the interior angles on one side is less than two right angles, then if the two lines are extended to a sufficient distance they will intersect each other on that side.

These postulates serve as axioms for Euclidean geometry, allowing proofs of many geometric properties and relations to be derived. The fifth axiom has been particularly important, giving rise to alternative geometries such as hyperbolic and elliptical geometry when modified.

circle

Peano's axioms

Peano's axioms are an axiomatic system for the natural numbers, consisting of the following key statements:

  1. The number zero is a natural number.
  2. Every natural number has a successor, which is also a natural number.
  3. Zero is not the successor of any natural number.
  4. Different numbers have different successors (injective nature).
  5. The property being true for zero and true for the successor of a natural number implies that this property is true for all natural numbers (induction principle).

These axioms define the fundamental properties of number and arithmetic, and form a framework from which the properties of addition and multiplication can be inferred.

0 Successor 1 Successor 2 Successor

Zermelo–Fraenkel set theory (ZF)

Set theory forms the basis of modern mathematical logic and is based on a set of Zermelo–Fraenkel axioms, often supplemented by the axiom of choice (AC), making it ZFC. These axioms are intended to formalize the properties of sets and include:

  • Axiom of Extensionality: Two sets are equal if their elements are equal.
  • Axiom of the empty set: There is a set that has no elements.
  • Axiom of Pairing: For any sets A and B, there is a set C whose elements are exactly A and B.
  • Axiom of Union: For a set of sets, there is a set that contains exactly the elements of those sets.
  • Axiom of Infinity: There exists a set containing 0 and the successor of each of its elements, which provides the basis for the construction of natural numbers.
  • Axiom of power set: For any set, there is a set of all its subsets.
  • Axiom of Regularity: Every nonempty set A has a member which is disjoint from A.

Set theory is an essential part of mathematical logic, enabling the comparison, construction, and manipulation of any mathematical object within the framework of sets.

Importance and applications of axiomatic systems

Axiomatic systems are important for several reasons:

  • Consistency: By starting with universally accepted axioms, mathematicians ensure that open mathematical systems are free from contradictions.
  • Clarity: Axioms provide a clear, fundamental foundation, promoting transparent and understandable progression toward complex theorems.
  • Universality: Theorems proved under an axiomatic system are universally valid within the scope of those axioms.

Axiomatic systems are used heavily in various mathematical fields, such as algebra and analysis, providing generalized structures and frameworks that can model and solve problems. This forms the basis of consistency in mathematical logic, which allows for advanced abstraction and synthesis of ideas within protected boundaries.

Developing a system of axioms

Creating an axiomatic system involves defining axioms with the following criteria:

  • Independence: The axioms must not be derivable from one another.
  • Consistency: New axioms should not contradict existing axioms in the system.
  • Completeness: The system must allow the derivation of every truth belonging to the mathematical context of the system.

For example, consider a hypothetical axiomatic system derived from three axioms:

  1. Axiom 1: All elements of the set are even numbers.
  2. Axiom 2: A set contains at least one number.
  3. Axiom 3: For every element x of a set, x+2 is also in the set.

From these axioms we can prove various theorems, such as every element is greater than or equal to the smallest element.

Challenges and limitations of axiomatic systems

Despite their wide utility and philosophical basis, axiomatic systems also face some limitations and challenges:

  • Incompleteness: Gödel's incompleteness theorem demonstrated inherent limitations in the formal axiomatic systems that can express arithmetic, and showed that no consistent system of axioms can be complete or capable of proving all truths.
  • Choice of axioms: Different initial axioms may lead to entirely different mathematical systems, making the choice of axioms difficult.
  • Explanation: Although axioms can in principle explain phenomena, their application may sometimes be subtle and open to different interpretations.

These limitations do not negate the importance of axiomatic systems; rather, they enrich our understanding by setting limits and challenges.

Conclusion

Axiomatic systems, which form the foundation of various mathematical disciplines, support the rigorous development and verification of mathematical theorems and concepts. From Euclidean geometry to modern set theory, axiomatic systems provide a systematic approach to unraveling and understanding the complex web of mathematical truths. Their importance lies not only in what they reveal, but also in how they challenge us to understand their limits, thereby inviting continual exploration and refinement in the world of mathematics.


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Axiomatic Systems

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