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


The study of "logic and foundations" is an important area in mathematics, especially at the PhD level. At its core, this subject is concerned with understanding the underlying principles that govern mathematical reasoning, ensuring that the structures we use in mathematics are consistent, solid, and complete. It involves exploring formal systems, set theory, model theory, and proof theory, among other areas.

Fundamentals of logic

Logic is essentially about formulating reasoned arguments and ensuring that the conclusions drawn are reasonable based on the premises given. At the heart of logical reasoning are statements or propositions, which are sentences that are either true or false, but not both. These propositions can be combined to form more complex logical structures using logical connectives such as:

  • AND (∧): A conjunction that is true if both propositions are true.
  • OR (∨): A disjunction that is true if at least one of the propositions is true.
  • Not (¬): A negation that reverses the truth value of a proposition.
  • Implies (→): A conditional that is false only if the first proposition is true and the second is false.
  • If and only if (↔): A biconditional that is true if both propositions are simultaneously true or false.

Formal logic involves using these basic connectives to construct valid arguments. Consider the propositions P and Q. We can form the following logical expression:

(P ∧ Q) → R

This statement is of the form "If both P and Q are true, then R is true." The validity of this expression can be found by listing all possible truth values of P, Q and R in a truth table and determining the truth value of the expression.

Truth tables

Truth tables are a useful visual tool for showing all possible values of logical expressions. Consider a simple example with two propositions:

P = It is raining
Q = The ground is wet

The expression P → Q (if it is raining, then the ground is wet) can have the following truth table:

| P | Q | P → Q |
|-----|-----|------------|
| T   | T   | T          |
| T   | F   | F          |
| F   | T   | T          |
| F   | F   | T          |

The only scenario where P → Q is false is when it is raining (P is true) but the ground is not wet (Q is false).

Set theory

At the foundation of modern mathematics, set theory serves as the framework for the construction of all mathematical objects. A set is simply a collection of distinct objects considered as a whole. For example, the set of natural numbers is represented as:

N = {0, 1, 2, 3, ...}

Sets can have the same special operations as numbers, including:

  • Union (∪): A set that contains all the elements of any set.
  • Intersection (∩): The set that contains all the elements that are common to both sets.
  • Difference (-): A set that has elements in one set but not in the other.
  • Complement (ˈ): All the elements that are not included in the set.

For example, let A and B be two sets:

A = {1, 2, 3}
B = {3, 4, 5}

The conjunction and intersection of A and B are as follows:

A ∪ B = {1, 2, 3, 4, 5}
A ∩ B = {3}

Visualizing these sets can help in understanding complex set operations.

A B 3

Foundations of mathematics

The foundational aspect of mathematics is deeply rooted in ensuring that the axioms or basic assumptions are inherently consistent and sufficient to derive all known mathematics. This aspect of logic and foundations is deeply influenced by work on formal systems, which attempt to formalize intuitive mathematical concepts.

Formal systems

A formal system consists of a formal language equipped with axioms and rules of inference. One of the earliest known formal systems is Euclidean geometry, based on Euclid's axioms. In a formal system, provided you start with the axioms, the subsequent use of inference rules will enable you to arrive at new truths within the system.

Take Peano's axioms, which provide a foundation for the natural numbers:

1. 0 is a number.
2. Every natural number has a successor, which is also a natural number.
3. 0 is not the successor of any natural number.
4. Different numbers have different successors.
5. A property that holds for 0 and holds for the successor of a number whenever it holds for that number, holds for all natural numbers.

The purpose of these axioms is to logically define the arithmetical properties of natural numbers. Using these axioms and logical reasoning, we can derive statements about numbers that we take to be certain.

Proof theory

Another essential field is proof theory, which studies the structure, types, and power of proofs. A proof is a logical argument that verifies the truth of a mathematical statement. In mathematics, there are different types of proofs, including:

  • Direct Proof: It involves the direct application of axioms and previous results to find the truth of a statement.
  • Indirect evidence: This often involves proving the opposite or proving by contradiction to establish the truth of a statement.
  • Constructive proof: It provides an explicit example or method to show that a mathematical object exists.

Example of direct evidence

An example of direct evidence may be related to even numbers.

Theorem: The sum of two even numbers is even.

Proof: Let m = 2a and n = 2b where a and b are integers. Then, m + n = 2a + 2b = 2(a + b), which is of the form 2k where k is an integer, thus m + n is even.

Model theory

Model theory deals with the relations between formal languages of logic and their interpretations or models. The model of a theory is a structure in which the sentences of the theory are true, providing a concrete context for interpreting abstractions.

A concrete mathematical system may have many models. For example, group theory can be applied to various structures such as addition, matrices, and numbers under rotation in geometry, demonstrating the generality of its application by satisfying the group axioms.

More text examples

Mathematical logic also introduces the concepts of consistency, completeness, and decidability.

  • Consistency: No contradiction can be derived from the axioms of a system. For example, if Peano's axioms are consistent then they cannot lead to a contradiction.
  • Completeness: Every statement in the language of the system can be either proven or disproved.
  • Decidability: There exists an effective method for determining whether a given statement is provable or refutable in the system.

Conclusion

The study of logic and foundation in mathematics delves deep into building rigorous systems and understanding the inner nature of mathematical structures and other mathematical branches. These fundamentals not only form coherent knowledge structures but also address philosophical questions related to mathematics. By mastering logic and foundation, research and general mathematical practice are taken to new levels, helping to draw accurate conclusions derived from solid foundations and clear reasoning.


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