Course.
Math 116b Winter 2018
1:00-2:25 TR, B127 GCL

Instructor.
Aristotelis Panagiotopoulos
Building 15, office 210-5
panagio at caltech.edu
Office hours. Mondays and Wednesdays 4-5 PM, Office 210-5, Building 15

Books. This class will be based on a complilation of book chapters and notes. It will therefore be important for each of you to have access to class notes.That being said, Enderton's "Elements of set theory" is going to be relevant for large parts of the quarter. For a more advanced book one may look at Kunen's "Set theory". Finally, a pleasant read for the last weeks of the quarter (or afterwards) might be Weaver's "Forcing for mathematicians".

Papers and extra material.

• A fun paper on cardinal arithmetic without axiom of choice.
• Intro to Hydra Game. Also here you can find an interesting survey on ordinal analysis for PA (and other theories)
• Seven useful properties of ordinals.
• A paper on Gödel's incompletness and finite combinatorics, see also Marker's "Model Theory" Chapter 5
• Homework.
HW1. (Solutions),
HW2. (Solutions revised with extra clarifications),
HW3. (Solutions),
HW4. (Solutions),
HW5. (Solutions),
HW6. (Solutions),
HW7. (Solutions).

Calendar.

• (Mar 8) Forcing and the continuum hypothesis II.
• (Mar 6) Forcing and the continuum hypothesis I.
• (Mar 1) Cardinal arithmetic II (the fundamental theorem); Godel well ordering; cofinality; regular and singular cardinals; inaccessible cardinals and a meta-theorem.
• (Feb 27) Order types and Cardinalities via Scott's trick; cardinals; Hartog's theorem; cardinal arithmetic I.
• (Feb 22) Axiom of regularity (equivalent forms); transitive closure, ranks, tree analysis; induction on well founded class relations.
• (Feb 20) Topology and normal functions on ORD; arithmetic operations on ORD; an interesting ordinal called ε0
• (Feb 15) Fundamental theorem of ordinals; more on classes ;induction and definition by induction on ORD; arithmetic operations on ORD.
• (Feb 13) Classes; class functions; Replacement axiom; ORD and V.
• (Feb 8) Zermelo's theorem; Ordinals; the limits of Zermelo's axioms.
• (Feb 6) Wellorderings; transfinite induction; comparing wellorderings.
• (Feb 1) Cantor Bendixson theorem; A game with two players; orderings.
• (Jan 30) The cardinality of the continuum and various properties; Continuum hypothesis; perfect sets.
• (Jan 25) Paris Harrington, a large cardinal principle; two proofs of Schröder Bernstein; Knaster-Tarski fixed point lemma.
• (Jan 23) Pigeonhole principle; trees; Konig's lemma; Infinite Ramsey theorem.
• (Jan 18) Axiom of choice and various notions of infinity; Countable sets, aleph-zero and properties; Cantor's powerset theorem.
• (Jan 16) Ordering on ω; Integers, Rationals, Reals; Infinite vs Dedekind infinite.
• (Jan 11) Construction of (ω,0,s,+,*); transitive sets; definition by recursion II.
• (Jan 9) Products, functions and other constructions from the axioms; number systems; definition by recursion.
• (Jan 4) Russel's Paradox; Zermelo's axioms and universe.

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