What is Mathematics: Gödel’s Theorem and Around

Written by: Karlis Podnieks (University of Latvia)

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.

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
5.2. 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