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## Around Goedel's TheoremHyper-textbook for students in
mathematical logic University of LatviaInstitute of
Mathematics and Computer Science |
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An extended translation of my book "Around
Goedel's theorem" published 1992 in Russian.**
**Copyright © 1997-2001 Karlis Podnieks. All
rights reserved.

**1. Platonism, intuition and the nature
of mathematics**

1.1. Platonism - the philosophy
of working mathematicians

1.2. Investigation of stable
models - the nature of the mathematical method

1.3. Intuition and axioms

1.4. Formal theories

1.5. Hilbert's program

**2. Axiomatic set theory**

2.1. 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
determinateness

2.4.4. Ackermann's
set theory (Church's thesis for set theory?)

**3. First order arithmetic**

3.1. From Peano axioms to first
order axioms

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 factorial function

4.7. Elimination of restricted
universal quantifiers

4.8. 30 ans apres

**5. Incompleteness theorems**

5.1. Liar's paradox

5.2. Self-reference lemma

5.3. Goedel's incompleteness
theorem

5.4. Goedel's second theorem

**6. Around Goedel'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

6.6. Loeb's theorem

**Appendix 1.** About model theory

**Appendix 2.** Around Ramsey's
theorem

**Appendix 3.** What is logic,
really?

**Appendix 4.** Descriptive set
theory

mathematics, logic, foundations, what is mathematics, incompleteness theorem, mathematical, Gödel, online, web, Godel, book, Goedel, tutorial, textbook, teaching, learning, study, student, Podnieks, Karlis, paradox, effectiveness, methodology, philosophy, formalism, Platonism, intuition, nature, theory, axiomatic, formal, Hilbert, program, twin prime conjecture, set theory, axiom, Zermelo, Fraenkel, Frankel, Cantor, Frege, Russell, Ramsey theorem, descriptive set theory, paradox, comprehension, infinity, continuum hypothesis, continuum problem, mathematical logic, constructibility, determinateness, descriptive, Ackermann, continuum, first order arithmetic, Peano, Dedekind, Grassmann, arithmetic, tenth problem, 10th, problem, Diophantine equation, Presburger, liar, self reference, theorem, Rosser, incompleteness, Ramsey, Russell paradox, liar paradox