Date
|
What we discussed/How we spent our time
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Jan 18
|
Snow day!
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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'.)
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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’.)
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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
-
Satisfaction, $\mathbb A\models \sigma$.
-
Semantic entailment, $\Sigma\models\sigma$.
-
Proof, $\Sigma\vdash\sigma$.
-
Soundness, Completeness, Decidability of proofs.
-
Gödel's Completeness Theorem.
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Feb 3
|
Reading on the axioms: LST 67-69, NST 78-80.
We discussed the
axioms of ZFC.
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Feb 6
|
Reading on the axioms: LST 67-69, NST 78-80.
We worked on the this handout.
We discussed
-
The goal of introducing all definitions
and theorems in a well-founded way.
-
ZFC with or without atoms/urelements.
-
The first few levels of the von Neumann universe.
-
Intersection versus union (we can form the union of any set
or any nonempty class).
-
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
-
Russell's Paradox.
-
Classes versus sets, proper classes.
-
The Kuratowski Encoding for ordered pairs. Tuples. Cartesian products.
-
Functions (+the Function Rule) and relations.
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Feb 10
|
Reading on elementary set theory: LST 70-74, NST 81-89.
We distributed these exercises and definitions.
We discussed
-
Terminology for functions.
-
Equipotence.
-
Finite and infinite sets.
-
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
-
The fact that $\langle \mathbb N; <\rangle$ is well-ordered.
-
The Recursion Theorem.
-
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
-
discussed the first homework assignment. (Due date for first draft, working groups, TeX resources.)
-
reviewed the Recursion Theorem and its refinements.
-
practiced some proofs of the Laws of Successor and the Laws of Addition on $\mathbb N$.
-
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.
- We discussed the enumeration induced
on a subset of an enumerated set.
-
We compared two proof sketches
that (WO) implies (Bases). One proof sketch was for
finitely generated spaces and the other was for arbitrary spaces.
- We discussed the enumeration of classes by ordinals,
including a brief discussion of the Axiom of Global Choice.
-
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.
- We discussed the enumeration induced
on a subset of an enumerated set.
-
We compared two proof sketches
that (WO) implies (Bases). One proof sketch was for
finitely generated spaces and the other was for arbitrary spaces.
- We discussed the enumeration of classes by ordinals,
including a brief discussion of the Axiom of Global Choice.
-
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.
- We defined cardinal sum, product, and exponentiation.
- We explained why the operations of
cardinal sum, product, and exponentiation are monotone operations.
- We explained why cardinal arithmetic
differs from ordinal arithmetic, but also why they
agree on finite cardinals.
- We proved that $\omega\cdot\omega=\omega$ (cardinal multiplication).
|
Mar 3
|
- We discussed the relationship between
Gödel's Completeness Theorem and
Gödel's First Incompleteness Theorem.
- We defined the Hartogs number of a set.
- 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.
- We discussed consequences of the Absorption Theorem
for cardinal arithmetic involving finite sums and products,
including $\kappa\cdot (\lambda+\mu)=\kappa\cdot\lambda+\kappa\cdot\mu$.
- 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
$\sum_{i<\lambda} \kappa_i =\lambda\cdot\sup\{\kappa_i\}$.
- 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.
- We discussed the motivation for the definition of
cofinal and coinitial, and defined
cofinality and coinitiality for any poset.
- We defined regular and singular
cardinals.
- We proved that successor cardinals are regular.
- 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.
- We reviewed cofinality. (Dominating and unbounded sets,
regular and singular cardinals.)
- We solved some challenge problems.
(What is the cofinality of the empty set?
Which natural numbers are regular cardinals?)
- 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.)
- We explained why $\textrm{cf}(\kappa)$ is a regular
cardinal.
- We proved that if $\kappa$ is an infinite cardinal, then
$\kappa<\kappa^{\textrm{cf}(\kappa)}\leq \kappa^{\kappa}=2^{\kappa}$.
- We introduced the
Gimel function: ג$(\kappa)=\kappa^{\textrm{cf}(\kappa)}$.
|
Mar 15
|
- We explained why $\textrm{cf}(2^{\lambda})>\lambda$
when $\lambda$ is infinite.
(In fact, $\textrm{cf}(\kappa^{\lambda})>\lambda$
when $\lambda$ is infinite and $\kappa\geq 2$.)
- We explained why $|\mathbb R|\neq \aleph_{\omega}$.
- 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$(\kappa)\leq 2^{\kappa}$.
- 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.
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May 1
|
We discussed the construction and properties of
$M[G]$.
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May 3
|
We practiced writing.
|