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 ↓


Proof Theory


Proof theory is an important branch of mathematical logic that penetrates to the core of mathematics: the concept of proof. It studies the structure, nature, and implications of mathematical proofs. By understanding proofs at a granular level, we can explore the foundations of mathematics and can provide insight into mathematical reasoning.

What is evidence theory?

At its core, proof theory is about understanding the formal aspects of logic. This involves the formalization of proofs, which are sequences of statements that lead from premises to conclusions. These sequences are governed by a set of rules that govern the basis of the conclusion. Determine the valid steps in drawing conclusions from a question.

Historical context

Proof theory arises from the broader context of foundational studies in mathematics. The motivation to formalize mathematics came from crises arising from paradoxes in set theory and concerns about the well-definedness of mathematical theories. A key figure, David Hilbert, proposed a method for formalizing mathematical theories using finite sets of mathematical concepts. Tried to establish consistency proofs for mathematics, which gave rise to proof theory as a formal discipline.

Formal systems and syntax

To understand proof theory, one must be familiar with formal systems. A formal system consists of:

  • Symbols: Basic units from which expressions are built.
  • Formula: A finite sequence of symbols constructed according to specific rules.
  • Axioms: assumed statements or starting points for constructing proofs.
  • Inference rules: Rules that guide valid transformations from one or more statements to others.

The syntax of a formal system deals with these structural aspects without concern for meaning or truth. This is similar to understanding the grammar of a language before giving meaning to its sentences.

An example of a formal system syntax

Symbols: {P, Q, ∧, ∨, ¬, →, (, )}

Formula: 
Example – (P ∧ Q), ¬P, (P → Q)

Axiom:
1. P ∨ ¬P (Law of Excluded Middle)

Inference rules:
Modus ponens: From P and (P → Q), infer Q.

Proofs as formal objects

In proof theory, proofs are viewed as formal objects, not just as tools for verifying statements. They are sequences or trees of formulas constructed using the axioms and inference rules of a formal system.

Visualization of evidence

Proofs can often be represented in tree-like structures where each node represents the application of an inference rule. Here's a simple example:

P p → q Why

In this diagram, starting with the premises P and P → Q, we derive the conclusion Q using the modus ponens rule.

Proof transformations and normal forms

An interesting aspect of proof theory is to figure out how different proofs of the same statement relate to each other. This involves investigating transformations such as removing unnecessary steps or converting to normal forms where the proof is organized in a standard way.

Consider any formula proven by contradiction. We can often turn such a proof into a constructive proof, which gives deeper insight into the nature of the formula.

Example of proof normalization

Suppose we initially have some complicated proof:

Hypotheses:
H1: ¬P ∨ Q
H2: P
H3: ¬Q

evidence:
1. P (from H2)
2. ¬(¬P) (by double negative on H2)
3. ¬Q (from H3)
4. ¬P ∨ Q (from H1)
5. Contradictions on steps 3 and 4

The essence of proof theory lies in simplifying and understanding such proofs. We can also show that by contradictions, we can always get P ∧ ¬P, which leads us nowhere unless it is transformed.

Types of arguments

Proof theory is not monotonous; it is based on a variety of logical systems. Some major types include:

  • Propositional logic: The simplest form, dealing with statements that are either true or false.
  • Predicate logic: extends propositional logic by including quantifiers and variables, allowing for more expressive statements.
  • Modal logic: examines necessity and possibility, involving operators such as "necessarily" and "possibly".

Cut elimination

A major result in proof theory, especially due to Gerhard Gentzen, is cut elimination. The cut rule allows the inclusion of intermediate assertions in proofs. The cut elimination theorem states that by simplifying (usually by lengthening) every proof such that which does not use the cut rule.

Cut elimination not only implies consistency but also leads to the axiom property, which means that every formula in a cut-free proof is an axiom of the original assumption or conclusion.

Illustration of cut elimination

Consider a proof framework:

Conclusion: A → B

Intermediate: A

Cut Rule Applications: 
If A ⊢ B and C ⊢ A, then find B from line A.

Through elimination, we modify the proof structure so that premises can be directly connected to conclusions without intermediate formulas.

Consistency and completeness

Proof theory provides tools for addressing key foundational questions such as consistency (no contradictions) and completeness (every true statement is provable).

Hilbert's program aimed to prove the consistency of mathematics, but Gödel's incompleteness theorems showed that no sufficiently expressive system could prove its own consistency.

A Brief Overview of Gödel's Incompleteness Theorem

First Theorem: Any consistent formal system that can express arithmetic cannot be complete.

Second Theorem: Such a system cannot prove its consistency.

These theorems reshaped the approach to proof theory and highlighted the limitations and strengths of formal systems.

Applications and future directions

Beyond basic mathematics, proof theory also has applications in computer science, particularly in areas such as programming language design and verification. Logical frameworks and proof assistants are based on proof-theoretic principles, which ensure that software is written in accordance with specifications. can be effectively verified against.

Future directions in proof theory may include further exploration of connections with computational complexity, the development of automated proof systems, and continued study of the philosophical implications of formal proofs.

Conclusion

Proof theory stands as a deep exploration of what lies at the core of mathematical logic. By examining the structures and transformations of proofs, it provides insights that reach beyond mathematics to fields such as logic, computer science, and philosophy. As we develop our understanding of proof theory, we get closer to understanding the full depth and breadth of human logical reasoning.


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