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Math 6730-001: Set Theory, Spring 2023


Lecture Topics


Date
What we discussed/How we spent our time
Jan 18
Snow day!

Jan 20
Course Notes
LST: Lectures on Set Theory by J. Donald Monk
NST: Notes on Set Theory by J. Donald Monk

Syllabus. Evaluation. Logic. Logic 2.

Jan 23
Reading on logic: LST 1-23, NST 1-29.

We continued to review first-order logic. (We stopped at 3(b) `atomic formulas'.)

Jan 25
Reading on logic: LST 1-23, NST 1-29.

We continued to review first-order logic. (We stopped at 4(b) ‘the effect of logical connectives on predicates’.)

Jan 27
Reading on logic: LST 1-23, NST 1-29.

We continued to review first-order logic. (We stopped at 4(c) `Logical equivalence. Prenex form'.)

Jan 30
We worked on practice problems (solutions).
Feb 1
Readings on proofs + completeness: LST 24-66, NST 30-77.

We discussed

  1. Satisfaction, $\mathbb A\models \sigma$.
  2. Semantic entailment, $\Sigma\models\sigma$.
  3. Proof, $\Sigma\vdash\sigma$.
  4. Soundness, Completeness, Decidability of proofs.
  5. Gödel's Completeness Theorem.
Feb 3
Reading on the axioms: LST 67-69, NST 78-80.

We discussed the axioms of ZFC.

Feb 6
Reading on the axioms: LST 67-69, NST 78-80.

We worked on the this handout. We discussed

  1. The goal of introducing all definitions and theorems in a well-founded way.
  2. ZFC with or without atoms/urelements.
  3. The first few levels of the von Neumann universe.
  4. Intersection versus union (we can form the union of any set or any nonempty class).
  5. The definition of the natural numbers as the least inductive set.
Feb 8
Reading on elementary set theory: LST 70-74, NST 81-89.

We distributed these exercises and these notes. We discussed

  1. Russell's Paradox.
  2. Classes versus sets, proper classes.
  3. The Kuratowski Encoding for ordered pairs. Tuples. Cartesian products.
  4. Functions (+the Function Rule) and relations.
Feb 10
Reading on elementary set theory: LST 70-74, NST 81-89.

We distributed these exercises and definitions. We discussed

  1. Terminology for functions.
  2. Equipotence.
  3. Finite and infinite sets.
  4. Properties of the natural numbers.
Feb 13
Reading on the Recursion Theorem: LST 80-86 or NST 95-103.

We distributed these exercises and definitions. We discussed

  1. The fact that $\langle \mathbb N; <\rangle$ is well-ordered.
  2. The Recursion Theorem.
  3. Refinements: Recursion with parameters, course-of-values recursion, and definition by recursion using well-founded set-like relations in place of $\langle \mathbb N; <\rangle$.
Feb 15
Reading on the Recursion Theorem: LST 80-86 or NST 95-103.

We distributed these hints to some of the exercises about the arithmetic of $\mathbb N$. We

  1. discussed the first homework assignment. (Due date for first draft, working groups, TeX resources.)
  2. reviewed the Recursion Theorem and its refinements.
  3. practiced some proofs of the Laws of Successor and the Laws of Addition on $\mathbb N$.
  4. discussed how the Axiom of Choice is used in conjunction with the Recursion Theorem to enumerate a finite set or to construct an injection $f\colon \mathbb N\to A$ for any infinite set $A$.
Feb 17
Reading on the Recursion Theorem: LST 75-79.

We discussed these notes on ordinals.

Feb 20
Reading on the Axiom of Choice: LST 106-110.

We discussed the equivalence of the Axiom of Choice and the Well-ordering Principle.

Feb 22
Reading on the Axiom of Choice: LST 91-110.

  1. We discussed the enumeration induced on a subset of an enumerated set.
  2. We compared two proof sketches that (WO) implies (Bases). One proof sketch was for finitely generated spaces and the other was for arbitrary spaces.
  3. We discussed the enumeration of classes by ordinals, including a brief discussion of the Axiom of Global Choice.
  4. We gave the definition of ordinal addition. We explained why $1+\omega= \omega$ while $\omega+1 = S(\omega)$, so $1+\omega\neq \omega+1$ is an instance of noncommutativity of ordinal addition.
Feb 24
Reading on Cardinals: LST 121-143.

  1. We discussed the enumeration induced on a subset of an enumerated set.
  2. We compared two proof sketches that (WO) implies (Bases). One proof sketch was for finitely generated spaces and the other was for arbitrary spaces.
  3. We discussed the enumeration of classes by ordinals, including a brief discussion of the Axiom of Global Choice.
  4. We gave the definition of ordinal addition. We explained why $1+\omega= \omega$ while $\omega+1 = S(\omega)$, so $1+\omega\neq \omega+1$ is an instance of noncommutativity of ordinal addition.
Feb 27
Reading on Cardinals: LST 121-143.

We began discussing this handout. We proved the Cantor-Schröder-Bernstein Theorem.

Mar 1
Reading on Cardinals: LST 121-143.

  1. We defined cardinal sum, product, and exponentiation.
  2. We explained why the operations of cardinal sum, product, and exponentiation are monotone operations.
  3. We explained why cardinal arithmetic differs from ordinal arithmetic, but also why they agree on finite cardinals.
  4. We proved that $\omega\cdot\omega=\omega$ (cardinal multiplication).
Mar 3
  1. We discussed the relationship between Gödel's Completeness Theorem and Gödel's First Incompleteness Theorem.
  2. We defined the Hartogs number of a set.
  3. We gave a ZF-proof that if $|A\times A|=|A|$ holds for every infinite set $A$, then the Axiom of Choice holds.
Mar 6
We reviewed Zorn‘s Lemma, and used it to prove in ZFC that every infinite set has a pairing function. From this we derived the Absorption Theorem for sums and products: if $0<\kappa, \lambda$, and either $\kappa$ or $\lambda$ is infinite, then $\kappa+\lambda=\kappa\cdot\lambda=\max(\kappa,\lambda)$.

At the end of the lecture we defined an amorphous set to be an infinite set whose subsets are finite or cofinite. We discussed that it is consistent with ZFC that amorphous sets exist. We explained why an amorphous set cannot have a pairing function.

Mar 8
Reading on Cardinals: LST 121-143.

  1. We discussed consequences of the Absorption Theorem for cardinal arithmetic involving finite sums and products, including $\kappa\cdot (\lambda+\mu)=\kappa\cdot\lambda+\kappa\cdot\mu$.
  2. We discussed some properties of cardinal arithmetic involving infinite sums and products, such as
    • $\kappa\cdot (\sum \lambda_i)=\sum \kappa\cdot \lambda_i$
    • $\sum_{i<\lambda} \kappa =\lambda\cdot\kappa$.
    • $\prod_{i<\lambda} \kappa =\kappa^{\lambda}$.
    • If each $\kappa_i$ is nonzero, then $\sum_{i<\lambda} \kappa_i =\lambda\cdot\sup\{\kappa_i\}$.
  3. We proved Kőnig‘s Theorem. We proved that Kőnig‘s Theorem implies the Axiom of Choice. We noted that Cantor‘s Theorem is a special case of Kőnig‘s Theorem. Conversely, it is not too hard to see that the proof of Kőnig‘s Theorem is a mild generalization of the proof of Cantor‘s Theorem.

Mar 10
Reading on Cofinality: LST 136-138.

  1. We discussed the motivation for the definition of cofinal and coinitial, and defined cofinality and coinitiality for any poset.
  2. We defined regular and singular cardinals.
  3. We proved that successor cardinals are regular.
  4. We noted that $\aleph_{\omega}$ is the first infinite singular cardinal, and that more generally $\aleph_{\alpha+\omega}$ is singular for any ordinal $\alpha$ since $\textrm{cf}(\aleph_{\alpha+\omega})=\aleph_0<\aleph_{\alpha+\omega}$.
Mar 13
Reading on Cofinality: LST 136-138.

  1. We reviewed cofinality. (Dominating and unbounded sets, regular and singular cardinals.)
  2. We solved some challenge problems. (What is the cofinality of the empty set? Which natural numbers are regular cardinals?)
  3. We proved the statement: If $\mathbb P$ is a total order, then $\mathbb P$ has a cofinal, increasing, $\textrm{cf}(\mathbb P)$-sequence. (This statement is equivalent to AC over ZF.)
  4. We explained why $\textrm{cf}(\kappa)$ is a regular cardinal.
  5. We proved that if $\kappa$ is an infinite cardinal, then $\kappa<\kappa^{\textrm{cf}(\kappa)}\leq \kappa^{\kappa}=2^{\kappa}$.
  6. We introduced the Gimel function: ג$(\kappa)=\kappa^{\textrm{cf}(\kappa)}$.
Mar 15
  1. We explained why $\textrm{cf}(2^{\lambda})>\lambda$ when $\lambda$ is infinite. (In fact, $\textrm{cf}(\kappa^{\lambda})>\lambda$ when $\lambda$ is infinite and $\kappa\geq 2$.)
  2. We explained why $|\mathbb R|\neq \aleph_{\omega}$.
  3. We (re)introduced the continuum function, the א-function, the ב-function, and the ג -function. Note that from Item (5) of the previous lecture we have $\kappa$ < ג$(\kappa)\leq 2^{\kappa}$.
  4. We proved most of the Main Theorem of Cardinal Arithmetic (Theorem 12.58 of LST). (We still need to prove case 3(b).)
Mar 17
We completed the proof of the MTCA. We discussed how it simplifies if we assume GCH, namely that
  • $\aleph_{\alpha}^{\aleph_{\beta}}=\aleph_{\beta+1}$ if $\aleph_{\alpha}\leq \aleph_{\beta}$,
  • $\aleph_{\alpha}^{\aleph_{\beta}}=\aleph_{\alpha+1}$ if $\textrm{cf}(\aleph_{\alpha})\leq\aleph_{\beta} <\aleph_{\alpha}$, and
  • $\aleph_{\alpha}^{\aleph_{\beta}}=\aleph_{\alpha}$ if $\aleph_{\beta}<\textrm{cf}(\aleph_{\alpha})$.
Mar 20
We discussed the process of creating algebraic models for computation. This led to a discussion of one way to show that a set of laws axiomatize a class of structures (= the Cayley Representation Theorem). We discussed sets of laws for the following classes:
  • Semigroups $\langle S; *\rangle$.
  • Semilattices $\langle S; \wedge\rangle$.
  • Bounded distributive lattices $\langle D; \wedge,\vee,0,1\rangle$.
  • Boolean algebras $\langle D; \wedge,\vee,\neg,0,1\rangle$.
Mar 22
We discussed this handout up to Remark (3) on page 2.
Mar 24
We reviewed the BA worksheet, explaining that
  • If $\langle S; \ast\rangle$ is a $\ast$-semilattice, then $(x\leq y)\leftrightarrow (x=x\ast y)$ defines a partial order on $S$ where $x\ast y$ equals the greatest lower bound of $x$ and $y$ with respect to $\leq$. Conversely, if $\langle S;\leq \rangle$ is a partial order in which any two elements have a glb, then $x\ast y:=\textrm{glb}(x,y)$ is a semilattice operation on $S$. These are inverse correspondences between semilattice operations on $S$ and partial orders under which any two elements have a glb. (Thus, a semilattice operation on $S$ carries the same information as a partial order on $S$ under which any two elements have a glb.)
  • We noted that any semilattice operation is monotone with respect to the semilattice order: $x_1\leq y_1$ and $x_2\leq y_2$ implies $x_1\ast x_2\leq y_1\ast y_2$.
  • We explained how absorption laws guarantee that the $\wedge$-order is dual to the $\vee$-order. We henceforth call the $\wedge$-order the ‘Boolean order’.
  • We explained why complemention is an anti-automorphism of any Boolean algebra.
  • We proved
    Thm. If $\mathbb B$ is a Boolean algebra and $X$ is the set of atoms of $\mathbb B$, then the function $$\varphi\colon \mathbb B\to \mathscr{P}(X)\colon b\mapsto \textrm{at}(b)$$ is a homomorphism. Here $\textrm{at}(b)=\{\alpha\in X\;|\;\alpha\leq b\}$ is the set of atoms of $\mathbb B$ below $b$. We claimed without proof that $\varphi$ is an isomorphism when $\mathbb B$ is finite. Later we will see that this map is an isomorphism whenever $\mathbb B$ is a complete atomic BA. Moreover, if we suitably generalize the concept of an atom and let $X$ be the set of generalized atoms, then we will see that the map $\varphi$ is always an embedding. This will complete the argument that our axiomatization of the laws of power set algebras is complete.
Apr 3
  • We showed that the homomorphism $\varphi$ in the theorem stated at the end of the last lectures notes is an isomorphism for finite BAs.
  • We defined what it means for a BA to be complete and to be atomic. We claimed that the homomorphism $\varphi$ from above is an isomorphism whenever $\mathbb B$ is a complete atomic BA.
  • The Lindenbaum-Tarski algebra for propositional logic, with an infinite set of propositional variables, is neither complete nor atomic. The BA FinCo$(X)$ of finite and cofinite subsets of an infinite set $X$ is atomic but not complete. The BA of regular open sets of the Cantor space is complete but not atomic.
  • We ended the lecture with the promise that we can better understand the structure of BAs if we generalize the concept of an atom.
Apr 5
We raised the question Was sind und was sollen die Atome? To answer this we introduced
  • ideals (proper, prime, maximal, principal).
  • filters (proper, ultra-, maximal, principal).
  • a complementary filter or ideal.
  • We proved that an ideal is prime iff it is maximal iff $(\forall b)((b\in I)\textrm{ or }(b’\in I))$. A dual result holds for filters.
  • We exhibited a nonprincipal ultrafilter in $\textrm{FinCo}(\omega)$.
  • We stated that if $\mathbb B$ is a BA and $X$ is the set of ultrafilters of $\mathbb B$, then the map $\varphi\colon \mathbb B\to {\mathcal P}(X)\colon b\mapsto \textrm{ult}(b)$ is an embedding.
Apr 7
We proved that if $\mathbb B$ is a BA and $X$ is the set of ultrafilters of $\mathbb B$, then the map $\varphi\colon \mathbb B\to {\mathcal P}(X)\colon b\mapsto \textrm{ult}(b)$ is an embedding. We worked from this handout.
Apr 10
Reading on rank: LST 161-166.

We discussed the set-theoretical hierarchy following this handout.

Apr 12
We discussed models of set theory following pages 3-5 of this handout.
Apr 14
We supplied some proofs for some of the results on pages 4-5 of this handout.
Apr 17
We completed the proof that if $V$ is a model of ZFC and $\gamma$ is inaccessible, then $V_{\gamma}$ is a model of ZFC.
Apr 19
We worked through this handout on forcing.
Apr 21
We started working through this handout on completions of forcing orders.
Apr 24
We discussed Galois connections and the embedding of of a forcing poset into its algebra of regular open sets,
Apr 26
We discussed Boolean-valued models these slides.
Apr 28
We continued discussing these slides.
May 1
We discussed the construction and properties of $M[G]$.
May 3
We practiced writing.