Course Page

Syllabus

Assignments

Important Dates

Reading

      Bibliography
      LST: Lectures on Set Theory by J. Donald Monk
      NST: Notes on Set Theory by J. Donald Monk
      MIL: A Mathematical Introduction to Logic by H. B. Enderton \(\require{AMSsymbols}\)

Jan 15
Syllabus
Background in Logic, pp. 1 - 4: Sentential logic
[Read: Lecture notes on Canvas;
LST or NST pp. 1 - mid 6.]

Jan 18
Martin Luther King Jr. Holiday. No class.

Jan 20
Background in Logic, pp. 5 - mid 10: First order logic for set theory (formulas, satisfaction)
[Read: Lecture notes on Canvas;
LST pp. 12 - mid 20; or NST pp. 16 - bottom 24.]

Jan 22
Background in Logic, pp. mid 10 - 13: First order logic for set theory (two notions of logical implication: $\models$ and $\models_\Box$, and their relationship; the logical axioms of a proof system for $\models$)
[Read: Lecture notes on Canvas;
LST pp. mid 20 - 22; or NST pp. bottom 24 - bottom 26;
MIL pp. 88 - 89, 109 - top 116.]

Jan 25
Background in Logic, pp. 14 - 18: First order logic for set theory (proof systems for $\models$ and $\models_\Box$; the relationship between the proof systems $\vdash$ and $\vdash_{\Box}$; metatheorems for $\vdash$).
[Read: Lecture notes on Canvas;
MIL Sections 2.4 - 2.5;
LST Sections 3 - 4; or NST Sections 3 - 4 (pp. 29 - 47, 49 - 63).]

Jan 27
The Axioms of Set Theory and Some Basic Consequences, pp. 1 - 5: The Axioms of Extensionality, Comprehension, Pairing, Union, Power Set, and some elementary consequences.
[Read: Lecture notes on Canvas;
LST Section 5 (pp. 67 - mid 68), Section 6 (p. 70); or
NST Section 5 (pp. 77 - mid 78), Section 6 (p. 80).]

Jan 29
The Axioms of Set Theory and Some Basic Consequences, pp. 6 - 10: Intersection and cartesian product; relations and functions. Classes, class relations, and class functions. The axioms of infinity, replacement, foundation, and choice.
[Read: Lecture notes on Canvas;
LST Section 5 (pp. mid 68 - 69), Section 6; or NST Section 5 (pp. mid 78 - 79), Section 6.]

Feb 1
Ordinals. Transfinite Induction and Recursion, pp. 1 - 4 (except Theorem 1.4 (ii)-(vi)): Definition of ordinals and some easy consequences. The relations $<$ and $\le$ on an ordinal and on the class $\mathbf{On}$ of all ordinals; < is a (strict) linear order.
[Read: Lecture notes on Canvas;
LST Section 7 (pp. 75 - mid 76); or NST Section 7 (pp. 89 - mid 90).]

Feb 3
Ordinals. Transfinite Induction and Recursion, pp. 3, 5 - 8: Properties of $<$ on ordinals and on $\mathbf{On}$. Ordinals are well-ordered by <. Successor ordinals and limit ordinals. The set $\omega$ of natural numbers is the smallest limit ordinal. Well-founded class relations and set-like class relations.
[Read: Lecture notes on Canvas;
LST Section 7 (pp. mid 76 - 78) and Section 8 (pp. 80 - mid 81); or
NST Section 7 (pp. mid 90 - 92) and Section 8 (pp. 94 - mid 95).]

Feb 5
Ordinals. Transfinite Induction and Recursion, pp. 9-10: Basic properties of well-founded and set-like class relations. Statement of the General Recursion Theorem.
[Read: Lecture notes on Canvas;
LST Section 8 (pp. mid 81 - mid 83, lower half of 85); or
NST Section 8 (pp. mid 95 - 97, lower half of 99).]

Feb 8
Ordinals. Transfinite Induction and Recursion, pp. 11 - 12: Proof of the General Recursion Theorem.
[Read: Lecture notes on Canvas;
LST Section 8 (pp. mid 83 - 86); or
NST Section 8 (pp. mid 97 - top 100).]

Feb 10
Ordinals. Transfinite Induction and Recursion, pp. 13 - mid 16: The transfinite induction and recursion theorems. Every well-order is isomorphic to $(\alpha,<)$ for a unique ordinal $\alpha$. Ordinal class functions.
[Read: Lecture notes on Canvas;
LST Section 9 (pp. 87 - top 90); or
NST Section 9 (pp. 103 - top 106).]

Feb 12
Ordinals. Transfinite Induction and Recursion, pp. mid 16 - 19: Basic properties of ordinal class functions. Ordinal arithmetic.
[Read: Lecture notes on Canvas;
LST Section 9 (pp. top 90 - 99); or
NST Section 9 (pp. top 106 - 115).]

Feb 15
The Axiom of Choice. Cardinals and Cardinal Arithmetic, pp. 1 - 3: Three statements that are equivalent in ZF to the Axiom of Choice: Choice Function Principle, Well-Ordering Principle, Zorn's Lemma.
[Read: Lecture notes on Canvas;
LST Section 10 (pp. 106 - mid 109); or
NST Section 10 (pp. 128 - mid 131).]

Feb 17
Wellness Day. No class.

Feb 19
The Axiom of Choice. Cardinals and Cardinal Arithmetic, pp. 4 - 7: Equipotence and cardinals. Finite and infinite sets. Cantor's Theorem. The ordinal class function $\aleph$. The cardinals form a proper class.
[Read: Lecture notes on Canvas;
LST Section 12 (pp. 121 - mid 125); or
NST Section 11 (pp. 136 - mid 140).]

Feb 22
The Axiom of Choice. Cardinals and Cardinal Arithmetic, pp. 8 - mid 13: Cardinal operations and their elementary properties. $\kappa\cdot\kappa=\kappa$ for infinite cardinals $\kappa$ (proof to be completed).
[Read: Lecture notes on Canvas;
LST Section 12 (pp. mid 125 - mid 136, except 12.32, 12.33, 12.35, 12.36, 12.41, 12.46); or
NST Section 11 (pp. mid 140 - mid 151, except 11.32, 11.33, 11.35, 11.36, 11.41, 11.46).]

Feb 24
The Axiom of Choice. Cardinals and Cardinal Arithmetic, pp. mid 13 - mid 16 (before Theorem 4.10): $\kappa\cdot\kappa=\kappa$ for infinite cardinals $\kappa$, and some consequences. König's Theorem. Cofinality.
[Read: Lecture notes on Canvas;
LST Section 12 (12.32, 12.33, 12.35, 12.36, 12.41, 12.46, and p. 136); or
NST Section 11 (11.32, 11.33, 11.35, 11.36, 11.41, 11.46, and p. 151).]

Feb 26
The Axiom of Choice. Cardinals and Cardinal Arithmetic, pp. 16 - 18 Cofinality. Regular and singular cardinals and their properties. $\text{cf}(\kappa)$ is a regular cardinal for all infinite cardinals $\kappa$. König's Theorem on cofinality.
[Read: Lecture notes on Canvas;
LST Section 12 (pp. 136 - 138); or
NST Section 11 (pp. 151 - 153).]

Mar 1
The Axiom of Choice. Cardinals and Cardinal Arithmetic, pp. 19 - top 22: The Main Theorem of Cardinal Arithmetic. Mathematics without the Axiom of Choice. Some weaker variants of AC.
[Read: Lecture notes on Canvas;
LST Section 12 (pp. 139 - 141); or
NST Section 11 (pp. 154 - mid 156).]

Mar 3
The Axiom of Choice. Cardinals and Cardinal Arithmetic, p. 22: Some weaker variants of AC (cont'd).
Trees, p. 1: Trees in set theory. König's Tree Lemma.
[Read: Lecture notes on Canvas;
LST Section 18 (p. 241); or
NST Section 22 (p. 369).]

Mar 5
Trees, pp. 2 - 4: $\kappa$-trees, $\kappa$-Aronszajn trees, and $\kappa$-Suslin trees. Existence of an $\omega_1$-Aronszajn tree.
[Read: Lecture notes on Canvas;
LST Section 18 (pp. bottom 241 - mid 244); or
NST Section 22 (pp. bottom 369 - 372).]

Mar 8
Trees, pp. bottom 4 - mid 6: Normal subtrees, eventually branching trees, and well-pruned $\kappa$-trees; examples and basic properties. Construction of a well-pruned normal $\kappa$-subtree of a $\kappa$-tree ($\kappa$ regular).
[Read: Lecture notes on Canvas;
LST Section 18 (pp. mid 244 - top 246); or
NST Section 22 (pp. 373 - mid 374).]

Mar 10
Trees, pp. mid 6 - 8: A characterization of $\kappa$-Suslin trees ($\kappa$ uncountable regular). Suslin lines and the Suslin Hypothesis. Idea of the proof of the equivalence (in ZFC) of the existence of a Suslin line and a Suslin tree.
[Read: Lecture notes on Canvas;
LST Section 18 (pp. top 246 - mid 251); or
NST Section 22 (Prop. 22.8, Thm. 22.14, Thm. 22.19).]

Mar 12
Clubs and Stationary Sets, pp. 1 - bottom 3: Definition and examples of clubs in ordinals. A characterization of clubs in limit ordinals, using normal functions. Clubs in regular cardinals. Existence of a club in $\alpha$ of order type cf$(\alpha)$.
[Read: Lecture notes on Canvas;
LST Section 19 (pp. 253 - bottom 254); or
NST Section 23 (pp. 405 - bottom 406).]

Mar 15
No class. Campus is closed, due to winter storm impacts.

Mar 17
Clubs and Stationary Sets, pp. bottom 3 - top 7: Intersections and diagonal intersections of clubs. Definition and some examples of stationary sets.
[Read: Lecture notes on Canvas;
LST Section 19 (pp. bottom 254 - mid 256); or
NST Section 23 (pp. bottom 406 - mid 408).]

Mar 19
Clubs and Stationary Sets, pp. top 7 - 8: Further examples of stationary sets. Fodor's Lemma.
[Read: Lecture notes on Canvas;
LST Section 19 (pp. mid 256 - mid 257); or
NST Section 23 (pp. mid 408 - mid 409).]

Mar 22
Clubs and Stationary Sets, pp. top 9: The combinatorial principle $\lozenge$;  $\lozenge$ implies CH.
Infinite Combinatorics, pp. 1 - 2: The general $\Delta$-system theorem.
[Read: Lecture notes on Canvas;
LST Section 19 (p. mid 257), Sections 16/20 (pp. mid 210 [def.] and 268); or
NST Section 23 (pp. mid 411), Sections 21/24 (pp. 356 - 356 [def] and 439).]

Mar 24
Infinite Combinatorics, pp. 3 - top 5, top 6: Applications and the indexed version of the general $\Delta$-system theorem. Partition calculus: arrow notation and some basic facts. Statement of Ramsey's theorem and its finite version.
[Read: Lecture notes on Canvas;
LST Section 20 (pp. bottom 268 - 269); or
NST Section 24 (pp. bottom 439 - 440).]

Mar 26
Infinite Combinatorics, pp. 5 - 8 (except Thm. 2.16): Proof of Ramsey's theorem. $2^\kappa\not\to(\kappa^+)^2_2$ for infinite cardinals $\kappa$. Dushnik--Miller and Erdős--Rado Theorems (proofs to be presented later). $2^\kappa\not\to (3)^2_\kappa$.
[Read: Lecture notes on Canvas;
LST Section 20 (pp. bottom 269 - 273); or
NST Section 24 (pp. mid 440 bottom 441) and Section 12 (pp. 186 - 187).]

Mar 29
Infinite Combinatorics, Thm. 2.16: $\omega\not\to(\omega)^\omega_2$.
Models of Set Theory, pp. 1 - 3: The set-theoretical hierarchy. Definition and properties of the rank of a set.
[Read: Lecture notes on Canvas;
LST Section 14 (pp. mid 161 - top 166); or
NST Section 12 (pp. 172 - mid 176).]

Mar 31
Models of Set Theory, pp. 4 - 6: Models of set theory. Checking the axioms. The axioms of ZFC, except the replacement axioms, hold in $V_\gamma$ if $\gamma$ is a limit ordinal $>\omega$. All axioms of ZFC hold in $V_\kappa$ if $\kappa$ is an uncountable strongly inaccessible cardinal.
[Read: Lecture notes on Canvas;
LST Section 14 (pp. 160 - mid 161, 166 - mid 168, and p. 178, starting in the 2nd paragraph); or
NST Section 14 (pp. 202 - mid 204, and Theorems 14.14, 14.16 on pp. 206 - 207).]

Apr 2
Models of Set Theory, pp. 7 - 9: Absoluteness. Absoluteness and $\Delta_0$-formulas, absoluteness for class functions.
[Read: Lecture notes on Canvas;
LST Section 14 (pp. mid 168 - mid 172); or
NST Section 13 (pp. 192 - 195).]

Apr 5
Models of Set Theory, pp. 10 - 13: Absoluteness upwards/downwards, absoluteness is preserved under composition. Absoluteness of formulas involving class functions. Applications to transitive class models of $\mathsf{ZF}$. Absoluteness of recursive definitions.
[Read: Lecture notes on Canvas;
LST Section 14 (pp. mid 172 - mid 176); or
NST Section 13 (pp. 196 - mid 199).]

Apr 7
Models of Set Theory, pp. 14 - top 17: Absoluteness of the rank function, and $V_\alpha^{\mathbf{M}}=V_\alpha\cap\mathbf{M}$ $(\alpha\in\mathbf{M})$ for transitive class models $\mathbf{M}$ of $\mathsf{ZF}$. The consistency of "There exist no inaccessible cardinals". The Mostowski collapse.
[Read: Lecture notes on Canvas;
LST Section 14 (pp. mid 176 - top 181); or
NST Section 13 (pp. mid 199 - mid 200), Section 16 (p. 215), Section 12 (pp. 180 - 181).]

Apr 9
Presentation by Connor Meredith: Dushnik - Miller Theorem.
Models of Set Theory, pp. top 17 - top 19: The Reflection Theorem.
[Read: Lecture notes on Canvas;
LST Section 14 (pp. top 181 - bottom 182); or
NST Section 15 (pp. mid 210 - bottom 211).]

Apr 12
Models of Set Theory, pp. top 19 - 21: Consequences of the Reflection Theorem. If $S$ is a consistent set of sentences containing $\mathsf{ZFC}$, then $S$ has a countable transitive model (c.t.m.).
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, p. 1: Introductory example.
[Read: Lecture notes on Canvas;
LST Section 14 (pp. bottom 182 - 184); or
NST Section 15 (pp. bottom 211 - mid 213).]

Apr 14
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, pp. 2 - top 6: Forcing orders and generic filters. Existence of $P$-generic filters for c.t.m.s of $\mathsf{ZFC}$. $P$-names; definition and basic properties. $G$-values of $P$-names; definition and basic properties. The definition and some properties of $M[G]$ for a c.t.m. $M$ of $\mathsf{ZFC}$.
[Read: Lecture notes on Canvas;
LST Section 15 (pp. 186 - top 187, top 188 - top 190); or
NST Section 27 (p. 533), Section 28 (pp. 562 - top 563, top 564 - top 566).]

Apr 16
Presentation by Ali Lotfi: $\Delta$-System Theorem.
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, pp. top 6 - mid 8: Further properties of $M[G]$ for a c.t.m. $M$ of $\mathsf{ZFC}$. If $\mathbb{P}\in M$ is a forcing order, and $G\not=\emptyset$ is a filter on $\mathbb{P}$, then $M$ and $M[G]$ have the same ordinals.
The idea of forcing. Definition of $\Vdash_{\mathbb{P},M}$.
[Read: Lecture notes on Canvas;
LST Section 15 (pp. top 190 - bottom 191, mid 192); or
NST Section 27 (p. 533), Section 28 (pp. top 566 - 567 [except 28.15-16], lower half of 568).]

Apr 19
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, pp. mid 8 - 12: The regular open sets of a topology form a complete Boolean algebra. The topology induced by a forcing order $\mathbb{P}=(P,\le 1)$, the Boolean algebra $\mathrm{RO}(\mathbb{P})$ of regular open sets, and its properties. The function $e\colon P\to\mathrm{RO}(\mathbb{P})$ and its properties.
Definition (by recursion) of the class functions $(\sigma,\tau)\mapsto[[\sigma=\tau]]$ and $(\sigma,\tau)\mapsto[[\sigma\in\tau]]$ that assign Boolean values (from $\mathrm{RO}(\mathbb{P})$) to atomic formulas and $P$-names $\sigma,\tau$ for their free variables. Extension of the definition to arbitrary formulas and $P$-names for their free variables.
[Read: Lecture notes on Canvas;
LST Section 13 (pp. 148 - top 155), Section 15 (pp. bottom 192, top 195); or
NST Section 27 (p. 533 - top 540), Section 28 (pp. 569 - mid 571).]

Apr 21
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, pp. 13 - mid 18: Definition of the forcing notion $\Vdash^*$ (in $\mathbf{V}$). The Forcing Theorem (with parts of the proof). The equivalence of $p\Vdash \varphi(\bar{\tau})$ and $\bigl(p\Vdash^*\varphi(\bar{\tau})\bigr)^M$.
Every generic extension $M[G]$ of a c.t.m. $M$ of $\mathsf{ZFC}$ is a c.t.m. of $\mathsf{ZFC}$ (proof to be continued).
[Read: Lecture notes on Canvas;
LST Section 15 (pp. top 195 - bottom 198); or
NST Section 28 (pp. mid 571 - bottom 574).]

Apr 23
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, pp. mid 18 - bottom 21: Every generic extension $M[G]$ of a c.t.m. $M$ of $\mathsf{ZFC}$ is a c.t.m. of $\mathsf{ZFC}$.
Preservation of cardinals: Given a c.t.m. $M$ of $\mathsf{ZFC}$, a forcing order $\mathbb{P}\in M$, and a cardinal $\kappa$ in $M$, the relationship between the conditions ``$\mathbb{P}$ preserves cardinals $\ge\kappa$'', ``$\mathbb{P}$ preserves cofinalities $\ge\kappa$'', and ``$\mathbb{P}$ preserves regular cardinals $\ge\kappa$''.
[Read: Lecture notes on Canvas;
LST Section 15 (pp. bottom 198 - mid 200), Section 16 (p. 208); or
NST Section 28 (pp. bottom 574 - mid 576), Section 29 (p. 587).]

Apr 26
Presentation by Sangman Lee: Erdős--Rado Theorem.
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, pp. bottom 21 - 22: The $\kappa$-chain condition for $\mathbb{P}$, and its application to 'capturing' functions $f\colon A\to B$ in $M[G]$ $(A,B\in M)$ by functions $F\colon A\to \mathcal{P}(B)$ in $M$ such that $f(a)\in F(a)$ and $(|F(a)|<\kappa)^M$ for all $a\in A$.
[Read: Lecture notes on Canvas;
LST Section 16 (p. 209); or
NST Section 29 (p. 588).]

Apr 28
Forcing. The consistency of $\mathsf{ZFC}+ \neg\mathrm{CH}$, pp. 23 - 26: If $\mathbb{P}\in M$ satisfies the $\kappa$-chain condition ($\kappa$ a regular cardinal in $M$), then $\mathbb{P}$ preserves cardinals $\ge\kappa$.
Cohen's Theorem and its corollary, proving that if $\mathsf{ZFC}$ is consistent, then so is $\mathsf{ZFC}+\neg\text{CH}$.
Brief sketch of how to use forcing to prove that if $\mathsf{ZFC}$ is consistent, then so is $\mathsf{ZFC}+\text{CH}$.
[Read: Lecture notes on Canvas;
LST Section 16 (pp. bottom 209 - top 212 and top 212 - mid 213); or
NST Section 29 (pp. 589, 586 - top 587, and bottom 589 - top 591).]

Homework

Assignment 1 (TEX file)
(Posted on Jan 30, first draft is due on Feb 8)
SOLUTIONS:     Problem 1     Problem 4     Problem 5     Problem 6     Problem 7

Assignment 2 (TEX file)
(Posted on Mar 8, first draft is due on Mar 17)
SOLUTIONS:     Problem 1     Problem 3     Problem 5     Problem 6     Problem 7

Assignment 3 (TEX file)
(Posted on Mar 31, first draft is due on Apr 12)
SOLUTIONS:     Problem 2     Problem 3     Problem 5     Problem 6     Problem 7

Presentations

The $\Delta$-System Theorem
Ali Lotfi, April 16

Dushnik–Miller Theorem
Connor Meredith, April 9.

Erdős–Rado Theorem
Sangman Lee, April 26