Lecture Topics for History Students

Math 4730: Introduction to Set Theory, Fall 2013

Lecture Topics

Date	What we discussed/How we spent our time
Aug 26	Syllabus. Text. Axioms. Some history about notation.
Aug 28	We discussed Axioms 1, 2, 4, 5, 6 and practiced writing them as formal sentences.
Aug 30	Axiom of Comprehension. Russell's Paradox. We showed that the intersection of a nonempty family of sets exists. We defined difference and symmetric difference of sets.
Sep 4	Ordered pairs. Cartesian product of sets. Relations. (Sections 2.1 and 2.2.)
Sep 6	Functions. (Section 2.3).
Sep 9	Review and Quiz 1.
Sep 11	Classes and class functions. Union, intersection and Cartesian product of systems of sets.
Sep 13	Class canceled due to flooding. Next Monday's quiz will be pushed back to Wednesday.
Sep 16	We discussed kernels and coimages of a functions, along with their abstractions: equivalence relations and partitions.
Sep 18	We described the 1-1 correspondence between equivalence relations and partitions. We defined strict and nonstrict partial and total orders, and mentioned how strict orders correspond to nonstrict orders. Quiz 2.
Sep 20	We introduced the successor function, inductive sets, the axiom of infinity, and the natural numbers. We examined the sets V₀, V₁, V₂, …, and defined the hereditarily finite sets.
Sep 23	We proved that the set, ω, of natural numbers is an inductive set, and that it is a subset of every inductive set. We showed that induction is a valid form of proof. We started on a proof that the relation "m∈n" is a strict total order on ω. Quiz 3.
Sep 25	We continued the proof that the relation "m∈n" is a strict total order on ω.
Sep 27	We finished the proof that the relation "m∈n" is a strict total order on ω. We showed that strong induction is a valid form of proof. We showed that if strong induction over an ordered set (X,<) is a valid form of proof, then (X,<) must be well founded.
Sep 30	We proved that strong induction over an ordered set (X,<) is a valid form of proof iff (X,<) is well founded. We discussed (informally) why an ordered set is well-founded iff it has no infinite descending chain. We defined well ordered sets and explained why "well order" = "well founded total order". We began discussing the arithmetic of the natural numbers. Quiz 4.
Oct 2	We discussed Quiz 4, in particular: (i) induction starting with base n=1 is valid, and (ii) formulas of the form (∀x P(x)) Q(x), which use restricted quantifiers, are abbreviations for conventional formulas, e.g. ∀x(P(x)→Q(x)). We then discussed definition by recursion, the recursive definition of x+y, x·y, and x^y, and proved some properties of arithmetic of natural numbers.
Oct 4	Proof of the recursion theorem.
Oct 7	Discussion of recursion, including the parametric version of recursion and the course of values version of recursion. Quiz 5.
Oct 9	Review for midterm.
Oct 11	Midterm.
Oct 14	Transitive closure of a binary relation. Axiom of Foundation: every set is well-founded under tr.cl(∈). Consequences: all sets are normal (i.e., x∈x is impossible), x∈y and y∈x cannot both hold, etc.
Oct 16	Cardinality. Cantor-Schröder-Bernstein Theorem.
Oct 18	Finite sets.
Oct 21	Countable and uncountable sets. Cantor's Theorem: \|A\| < \|P(A)\|. Quiz 6.
Oct 23	We showed that \|2^ω\|=\|(0,1)\|=\|ℝ\|. We showed that an infinite subset of a countably infinite set is countably infinite. We stated the Big Countability Theorem: if A is an alphabet that is at most countable, A^* is the set of finite sequences of element of A, W is a subset of A^*, and s:W→B is is surjective, then B is at most countable. (Hence, if the elements of B have finite-length descriptions in some countable alphabet, then B is countable.)
Oct 25	We proved the Big Countability Theorem.
Oct 28	We defined ordinal and cardinal numbers, and gave the intuition for them. We began a discussion of well-ordered sets. Quiz 7.
Oct 30	In response to a question, we discussed why \|(2^ω)^ω\|= \|2^ω×ω\|. Then we discussed isomorphism of ordered sets, and proved Lm 6.1.4, Cor 6.1.5 and part of Thm 6.1.3.
Nov 1	We finished the proof of Thm 6.1.3 and started discussing ordinals.
Nov 4	We showed that ordinal α is a subset of ordinal β iff α=β or α∈β. Then we showed that the class On of ordinal numbers is transitive and well-ordered by ∈ (in the sense that any nonempty subclass has a least element). We discussed the Burali-Forti Paradox. Quiz 8.
Nov 6	Order type of a well-ordered set. Transfinite recursion and induction.
Nov 8	Ordinal arithmetic.
Nov 11	We proved some results by transfinite induction, namely: Lm. δ<γ ↔ β+δ<β+γ, and Thm. (α+β)+γ=α+(β+γ) Quiz 9.
Nov 13	We discussed writing ordinals in "base α" for an ordinal α>1. Cantor normal form = base ω. We discussed how to compare numbers and how to add them if they are written in Cantor normal form.
Nov 15	Initial ordinals. Cardinals numbers. A set is equipotent with a cardinal number iff it can be well-ordered. Hartogs' Theorem.
Nov 18	The Hartogs number of a set is an initial ordinal. Successor versus limit cardinals. Definition of the arithmetic operations on cardinals. The ℵ-sequence versus the ℶ-sequence. Quiz 10.
Nov 20	Laws of cardinal arithmetic, including ℵ_α· ℵ_α= ℵ_α. Corollary: If at least one of the cardinals κ and λ is infinite and neither is zero, then κ+λ= κ·λ= max(κ,λ).
Nov 22	Axiom of choice.
Dec 2	We proved that AC is equivalent to several similar statements (e.g., (i) every partition has a transversal and (ii) every surjection has a section). We also showed that AC implies that every set can be well ordered. No quiz!
Dec 4	We proved in ZF that AC (axiom of choice), WO (well ordering principle) and ZL (Zorn's Lemma) are equivalent.
Dec 6	Every vector space has a basis (a proof via WO and a proof via ZL).
Dec 9	Chapter 9: Definition of infinite sum and product of cardinals. Thm. If λ is infinite and κ_α> 0 for all α<λ, then ∑_(α<λ)κ_α= λ·sup(κ_α). Thm. If λ is infinite, κ_α> 0 for all α<λ, and the sequence of κ's is increasing, then ∏_(α<λ)κ_α= sup(κ_α)^λ. König's Thm. If λ_α<κ_α for all α<μ, then ∑_(α<μ) λ_α < ∏_(α<μ) κ_α. Cor 1. κ<2^κ. Cor 2. \|ℝ\| = 2^ℵ₀ ≠ ℵ_ω.
Dec 9	Chapter 9: Definition of the continuum function. CH and GCH. Cofinality. Regular and singular cardinals. A cardinal has cofinality at most λ iff it is a sum of at most λ smaller cardinals. ℵ_α < cf(2^ℵ_α). Statement of Easton's Theorem.