A list of interesting topics I will read when I have spare time.

Ich muss wissen. Ich will weiss.

Logic
• Gaps between 1st order logic and 2nd order logic. (See  and here)
• Compactness theorem does not hold: there is a countable but unsatisfiable set of 2nd order logic sentences, every finite subset of which is satisfiable.
• Löwenheim-Skolem theorem does not hold: there is a 2nd order logic sentence whose models are exactly those with noncountable universes.
• There is a 2nd order logic sentence whose only model (up to isomorphism) is N.
• There is a 2nd order truth definition for 1st order sentences in N.
• Reachability and Hamiltonian Path is definable in 2nd order logic. (See )
• Only 1st order logic satisfies 0-1 law.
• Why the name Compactness theorem? What does it relate to compactness in topology?
• Intuionistic 1st order logic (See )
• Infinitary logic
• Modal logic
• Lindström's theorem for characterizing 1st order logic as maximal logic admitting compactness theorem and Löwenheim-Skolem theorem (See   pp. 261)
• Gentzen's proof of consistency of number theory in infinitary logic (See )
• Fixed-points
• Rosser's method to construct 1st-order undecidable sentences (See )
• Natural undecidable 1st-order sentences (e.g. Extended finite Ramsey theorem and Goodstein sequences. Also see here) (See )
• Hilbert's 10th problem: the impossibility of obtaining a general solution to Diophantine equations (See    and here)
Interesting/surprising consequences:
• There is a polynomial P such that nonnegative values of P form the set of prime numbers.
• More generally, so long as the set is r.e., there is a polynomial enumerating it.
• There is a single polynomial such that its zero exists iff there are positive integers x,y,z,n (n > 2) satisfying xn+yn=zn
• There is a "universal" polynomial U(y,x1,...,x9) such that for any polynomial P, P has zeros iff U(n,x1,..,x9) has zeros (for some n).
• J.P. Jones published many papers on Diophantine equations, see here
• Restricted version of Hilbert's 10th problem
Some known results:
• Diophantine equations with 9 variables are unsolvable (if solutions in N are sought), and with 21 variables are unsolvable (if solutions in Z are sought)
• Diophantine equations with degree 4 are unsolvable.
• Diophantine equations with degree 2 are solvable, but the algorithm does not run in polynomial time (i.e. the problem is NP-complete)
• Decidable fragments of number theory (See )
• Translation of number theory in ZFC
• Other implications of Incompleteness theorem (See )
• More applications of self-application (See )
• Decidability/Undecidability of first-order logic theories:
• Theory of abelian groups (Algebraically Closed Fields and Real Closed Fields) are decidable (because they are complete) (See )
• Theory of groups (fields, ordered fields) is undecidable.
• The universal theory of groups (i.e. word problem for groups) is undecidable. (due to Novikov and Boone. See  and )
• Techniques to prove decidability:
(See , Chapter C.3 and here)
• Quantifier elimination (Presburger used it to prove Theory of N with Addition. Tarski used it to prove Theory of R with Addition and Multiplication.) (See  pp. 595)
Also see below for computational complexity of decidable logic theories.
• Model-theoretic methods such as Vaught's test and categoricity (Cantor used it to prove Theory of Dense Linearly Ordered set)
Recursive Function Theory
• Degrees and jump operator.
• Creative sets, productive sets, simple sets, immune sets (see  )
• Arithmetic hierarchy (see  )
• Hyperarithmetic hierarchy (see  )
• Analytical hierarchy (see  )
• Grzegorzyk hierarchy (see )
• Post's problem and Friedberg's solution by Finite Injury Priority Method (see   )
• Admissible ordinals and alpha-recursion (See  pp. 653)
• Other computation models: functionals(See ), lambda calculus, Post machine, Markov algorithms, type-0 grammar, rewriting systems
Computational Complexity
• Limitation of formal systems to prove randomness (Chaitin's algorithmic information theory)
• Complexity of logic theories
• Time complexity of WS1S (weakly monadic 2nd order theory of successor.) Monadic means predicates can only take one argument. 1 means there is only 1 predicate. Weak means the underlying set is finite.
• S1S is equivalent to regular expressions and is thus decidable (proved by Büchi in "Weak Second-Order Arithmetic and Finite Automata")
• SnS are also decidable (see Rabin's "Decidability of Second-Order Theories and Automata on Infinite Trees" and TW's "Generalized Finite Automata Theory with an Application to a Decision Problem"), but the time complexity is non-elementary (i.e. stack of exponentials.)
• Time complexity of Presburger arithmetic
• Time complexity of Real Arithmetic
• Characterization of PH by "alternation"
• Approximability (See  )
• Non-approximability, APX, PTAS, FPTAS
• Approximation preserving reductions
• PCP Theorem: NP=PCP[O(log n),O(1)]
• Applications of list decoding in complexity theory
• Non-approximability via PCP
• Unrelativizable results, e.g. algebraic methods (arithmetization) that proves IP=PSPACE
• Boolean circuit complexity
• Berman-Hartmanis conjecture
• Ackermann function is not primitive recursive
• Kolmogorov complexity
Large Cardinal Theory
• Inaccessible cardinals
• Mahlo cardinals
• Weakly/Strongly Compact cardinals
• Ulam's measurable cardinals
• Lebesgue's measure problem: Does there exist a countably additive, invariant under translation and nontrivial (i.e. points have measure 0, and not all subsets have measure 0) measure defined on all subsets of R?
Ulam proved that any set possessing such a measure is of inaccessible cardinal. If we add the large cardinal axiom to ZFC, then the new theory is consistent with such a measure.
Banach and Kuratowski proved Continuum Hypothesis implies R is not real-valued measurable.
• Transfinite induction (e.g. used to prove Goodstein's Theorem)
Axiomatic Set Theory
• Zermelo-Fraenkel set theory
• von Neumann-Bernays-Gödel set theory
• Axiom of Choice , its consequences (e.g. Banach-Tarski paradox, also see here and here, ,), and its equivalences (e.g. Zorn's lemma )
• The Reflection Principle (Löwenheim-Skolem theorem ?)
• Constructible universe L, cumulative hierarchy V, and Axiom of Constructibility "V=L" (See )
• Fine structure of the constructible universe by Jensen
• Determinacy axioms
• Inner model
• Quine's New Foundation
• Forcing (Cohen's method and Boolean-valued Models)
• Independence results: Continuum Hypothesis (via generic sets) , Axiom of Choice (via urelements), Souslin's Hypothesis, Parallel Postulate
• Martin's axiom
• Aczel's Anti-Foundation Axiom (See )
Model Theory
• Saturated models and atomic models.
• Ultraproducts and ultrafilters
• Omitting types
• Model completeness
• Ehrenfeucht-Fraisse games
• Finite structures and 0-1 law
• Morley's categoricity theorem (simply put, if countable theory T is kappa-categorical, where kappa is uncountable, then T is lambda-categorical for every uncountable lambda)
• Vaught's conjecture and Knight's counter-example
• No complete theory has exactly 2 countable models (Vaught's theorem) (see )
• A complete theory with exactly n (n > 2) countable models (see )
• A complete theory with countably many nonisomorphic countable models (see )
• A complete theory with uncountably many nonisomorphic countable models (proof)
• Nonstandard models for N,Q, and R, and Nonstandard Analysis
• Stability, simplicity, forking
Category and Topos
Stochastic Processes
• Measure-theoretic probability
• High dimensional random walks (the probability of returning to origin is 1 in 1-D and 2-D random walks, but < 1 in higher-dimensions. This was proved by Pólya)
• Brownian motions, Wiener processes, Levy processes
• Martingales
• Kalman filter
• Diffusion equations
• Ito integration
• Stochastic calculus, stochastic differential equations
• Complexification and Karhunen-Loeve expansion
• Queueing theory with heavy-traffic, diffusion approximation
Phase Transitions and Critical Phenomena
• Mean-field theory
• Ehrenfest's classification of phase transitions
• Widom & Kadanoff scaling hypothesis
• Order parameters
• Landau-Ginzburg theory
• Renormalization group
• Ising models and Onsager's solution
• Long-range order, correlation length
• Continuous symmetries
• Self-organized criticality
• Percolation
• Catastrophe theory, Morse theory
Low-temperature Physics and Superfluidity
• The lambda point (the critical temperature: 2.17 K)
• Landau's two-fluid model of liquid helium-4 II
• Superfluidity of helium-3
• Bose-Einstein condensation

References
1. Handbook of Mathematical Logic (edited by Barwise)
2. Aspects of Incompleteness (by Lindström)
3. Computability and Logic (by Boolos and Jeffrey)
4. Gems of Theoretical Computer Science (by Schöning and Pruim)
5. The Incompleteness Phenomenon (by Goldstern and Judah)
6. Model Theory (3rd edition, by Chang and Keisler)
7. Theory of Recursive Functions and Effective Computability (by Hartley Rogers, Jr)
8. Set Theory (by Jech)
9. Set Theory - An Introduction to Independence Proofs (by Kunen)
10. Introduction to Lattices and Order (by Davey and Priestley)
11. Recursive Function Theory and Logic (by Yasuhara)
12. The Classical Decision Problem (by Börger, Grädel, and Gurevich)
13. A Course in Mathematical Logic (by Bell and Machover)
14. Set Theory and the Continuum Hypothesis (by Cohen)
15. Introduction to Set Theory (by Hrbacek and Jech)
16. Recursively Enumerable Sets and Degrees (by Soare)
17. Infinity and the Mind (by Rucker)
18. Cornerstones of Undecidability (by Rozenberg and Salomaa)
19. Mathematical Logic (by Shoenfield)
20. Diagonalization and Self-Reference (by Smullyan)
21. The Incompleteness Phenomenon: A New Course in Mathematical Logic (by Goldstern and Judah)
22. A Shorter Model Theory (by Wilfrid Hodges)
23. A Mathematical Introduction to Logic (by Enderton)
24. Mathematical Logic (by Ebbinghaus, Flum, and Thomas)
25. Computational Complexity (by Papadimitriou)
26. Proof Theory (by Takeuti)
27. Hilbert's Tenth Problem (by Matiyasevich)
28. Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (by Ausiello and et al.)
29. The Complexity Theory Companion (by Hemaspaandra & Ogihara)
30. Constructibility (by Devlin)
31. Vicious Circles (by Barwise & Moss)
32. Approximation Algorithms (by V.V. Vazirani)
33. The Banach-Tarski Paradox (by Wagon)
34. Complex Analysis (by Ahlfors)
35. Functions of One Complex Variable II (by J. B. Conway)
36. Classical Topology and Combinatorial Group Theory, 2nd edition (by J. Stillwell)
37. The Theory of Groups, 2nd edition (by J. Rotman)
38. The Pea and the Sun: A Mathematical Paradox (by Wapner)