Free advanced mathematics online hyper-textbook for students from University of Latvia professor Karlis Podnieks. An earlier version of the text is available in the original Russian.
Table of Contents
- 1. Platonism, intuition and the nature of mathematics
- 1.1. Platonism – the Philosophy of Working Mathematicians
- 1.2. Investigation of Stable Self-contained Models – the True Nature of the Mathematical Method
- 1.3. Intuition and Axioms
- 1.4. Formal Theories
- 1.5. Hilbert’s Program
- 1.6. Some Replies to Critics
- 2. Axiomatic Set Theory
- 2.1. The Origin of Cantor’s Set Theory
- 2.2 Formalization of Cantor’s Inconsistent Set Theory
- 2.3. Zermelo-Fraenkel Axioms
- 2.4. Around the Continuum Problem
- 2.4.1. Counting Infinite Sets
- 2.4.2. Axiom of Constructibility
- 2.4.3. Axiom of Determinacy
- 2.4.4. Ackermann’s Set Theory (Church’s Thesis for Set Theory?)
- 2.4.5. Large Cardinal Axioms
- 3. First Order Arithmetic
- 3.1. From Peano Axioms to First Order Arithmetic
- 3.2. How to Find Arithmetic in Other Formal Theories
- 3.3. Representation Theorem
- 4. Hilbert’s Tenth Problem
- 4.1. History of the Problem. Story of the Solution
- 4.2. Plan of the Proof
- 4.3. Investigation of Fermat’s Equation
- 4.4. Diophantine Representation of Solutions of Fermat’s Equation
- 4.5. Diophantine Representation of the Exponential Function
- 4.6. Diophantine Representation of Binomial Coefficients and the Factorial Function
- 4.7. Elimination of Restricted Universal Quantifiers
- 4.8. 30 Ans Apres
- 5. Incompleteness Theorems
- 5.1. Liar’s Paradox
- 5.2. Arithmetization and Self-Reference Lemma
- 5.3. Gödel’s Incompleteness Theorem
- 5.4. Gödel’s Second Incompleteness Theorem
- 6. Around Gödel’s Theorem
- 6.1. Methodological Consequences
- 6.2. Double Incompleteness Theorem
- 6.3. Is Mathematics “Creative”?
- 6.4. On the Size of Proofs
- 6.5. Diophantine Incompleteness Theorem: Natural Number System Evolving?
- 6.6. Löb’s Theorem
- 6.7. Consistent Universal Statements Are Provable
- 6.8. Berry’s Paradox and Incompleteness. Chaitin’s Theorem
- Appendix 1. About Model Theory
- Appendix 2. Around Ramsey’s Theorem
View this Free Online Material at the source:
What is Mathematics: Gödel’s Theorem and Around
