Date
|
What we discussed/How we spent our time
|
Aug 28
|
Syllabus. Text.
Some discussion about the foundations of mathematics.
The idea of a set. The axioms.
The directed graph model of set theory.
We explained the meaning of the axioms of
Extensionality, Empty Set, Infinity,
Pairing, and Union.
|
Aug 30
|
We discussed the language of set theory.
We wrote some axioms in a formal way.
We introduced the Directed Graph Model of set theory.
|
Sep 1
|
We discussed how to enlarge our language through definitions,
and we introduced definitions for $\emptyset, \subseteq$,
and ${\mathcal P}(X)$.
We discussed Axioms 1, 3, 4, 5, and 6 (i) in English, (ii)
in terms of Venn diagrams and examples, (iii) in terms
of the Directed Graph Model of set theory,
and (iv) in the formal language of Set Theory.
|
Sep 6
|
We took a
practice quiz.
Then we introduced the successor operation $S(x)=x\cup \{x\}$
and used it to define inductive sets.
We stated the Axiom of Infinity.
|
Sep 8
|
We discussed the Axiom of Separation.
We discussed Unrestricted Comprehension versus Restricted Comprehension.
We explained why naive set theory is inconsistent
and explained how naive set
theory differs from axiomatic set theory .
We explained
the difference between sets and classes,
and discussed some examples of proper classes.
(The Russell class, the class of all sets,
the class of all $1$-element sets.)
We began (but did not complete)
a discussion of the last three axioms.
|
Sep 11
|
We completed the introduction of the ZFC axioms
by discussing the last three axioms
(Replacement, Choice, Foundation).
We explained why we don't need an axiom
to discuss intersections ($\bigcap A$ can be constructed
using the Axiom of Separation).
Our discussion ended with me posing the following
questions:
"What could $\bigcup A$ or $\bigcap A$
mean if $A=\emptyset$?"
"What could $\bigcup A$ or $\bigcap A$
mean if $A$ is a class?"
Quiz 1!
|
Sep 13
|
We discussed union and intersection
in the following extreme cases:
the $\bigcup/\bigcap$ of a class $C$ that (i) has
one element, (ii) has zero elements, (iii)
is a proper class.
We defined $\mathbb N$ as the intersection of all
inductive sets. We explained why $\mathbb N$ is inductive
and why it is the least inductive set.
We discussed the first three pages of these
notes on Ordered Pairs and Relations.
|
Sep 15
|
We discussed pages 3-9 of these
notes on Ordered Pairs and Relations.
|
Sep 18
|
We discussed terminology for binary relations. The main topics were:
Directed graph representations of binary relations.
Order relations (e.g., $\leq$ on $\mathbb R$, or $\subseteq$
on ${\mathcal P}(X)$)
- Partial orders (reflexive, antisymmetric, transitive relations)
- Total orders (partial orders satisfying the law of trichotomy)
- Strict partial orders (irreflexive, transitive relations)
- Strict total orders (strict partial orders satisfying the law of trichotomy)
Quiz 2!
|
Sep 20
|
Today we defined
the kernel of a function.
(If $F\colon A\to B$ is a function, then
$\ker(F)=\{(a,a')\in A\times A\;|\;F(a)=F(a')\}$.)
Equivalence relation.
(= a reflexive, symmetric, transitive binary relation.)
equivalence class ($[a]=[a]_E=\{b\in A\;|\;(a,b)\in E\}$)
the set of $A/E=\{[a]_E\;|\;a\in A\}$ of $E$-equivalence classes.
We read $A/E$ as ``$A$ modulo $E$'' or ``$A$ mod $E$''.
natural map associated to an equivalence relation ($\nu(a)=[a]_E$)
We proved that a binary relation $E\subseteq A\times A$
is a kernel of a function if and only if it is
an equivalence relation. The proof showed that any
equivalence relation $E$ on a set $A$ is the kernel of
the natural map associated to $E$.
|
Sep 22
|
Today we discussed this handout
about function terminology. We spent some time relating
the four concepts:
kernel of a function/equivalence relation
coimage of a function/partition
The main elements of the relationship are these:
The kernel of a function is an equivalence relation.
Any equivalence relation is the kernel of a function.
Hence the definition of ``equivalence relation''
correctly axiomatizes kernels.
The coimage of a function is a partition.
Any partition is the coimage of a function.
Hence the definition of ``partition''
correctly axiomatizes coimages.
Equivalence relations on $A$ and partitions of $A$
``carry the same information''.
The main difference is that equivalence relations
are sets of pairs, while partitions are sets of sets.
|
Sep 25
|
We reviewed function terminology,
and then started discussing
these slides
on induction and recursion.
Quiz 3!
|
Sep 27
|
We finished
these slides
on induction and recursion and started on
these slides
on the order on $\mathbb N$.
|
Sep 29
|
We finished
these slides
on the order on $\mathbb N$ and started discussing
the arithmetic of $\mathbb N$.
Review sheet for the midterm.
|
Oct 2
|
We worked in small groups to
prove all of the statements in
this handout
about the arithmetic of $\mathbb N$.
Review sheet for the midterm.
Quiz 4!
|
Oct 4
|
We reviewed for the midterm following
this review sheet.
(The midterm will be held in class on October 6.)
We discussed this handout
on how to answer a question.
|
Oct 6
|
Midterm!
Midterm Answer Key.
|
Oct 9
|
We began to discuss cardinality
following this handout
and these slides.
|
Oct 11
|
Today we discussed
the definitions of
finite and infinite, the Pigeonhole Principle,
and we explained why $\mathbb N$ is infinite.
In the last 10 minutes we proved the
Cantor-Schröder-Bernstein Theorem.
|
Oct 13
|
We showed that ``apple addition''
satisfies the recursive definition
of addition, so all of the
arithmetical properties we proved
for addition
hold for apple addition.
We defined the characteristic function of a subset
and the support of a characteristic function.
The maps (subset $\mapsto$ its characteristic function)
and (characteristic function $\mapsto$ its support)
are inverse bijections between
${\mathcal P}(A)$ and $2^A$.
|
Oct 16
|
We proved Cantor's Theorem.
Quiz 5!
|
Oct 18
|
We explain why $|\mathbb R| = |\textrm{Cantor set}| = |(0_{\mathbb R},1_{\mathbb R})| =
|{\mathcal P}(\mathbb N)|=2^{\aleph_0}$.
The we showed that each of the following sets also have cardinality
$2^{\aleph_0}$:
The set of all functions $f\colon \mathbb N\to \mathbb N$.
The set of all injective functions $f\colon \mathbb N\to \mathbb N$.
The set of all surjective functions $f\colon \mathbb N\to \mathbb N$.
The set of all bijective functions $f\colon \mathbb N\to \mathbb N$.
|
Oct 20
|
We introduced `amorphous sets' as
we discussed the relationship between infinite
sets and Dedekind infinite sets. (Amorphous sets
are infinite but not Dedekind infinite.)
|
Oct 23
|
We completed these slides
on Dedekind infinite sets.
Part of one proof used a modification of
the Axiom of Choice, so we started
discussing slight modifications
of AC. Namely, we noted that ZF+AC is equivalent
to ZF + any of these statements:
every partition has a transversal.
every surjective function has a section.
Quiz 6!
|
Oct 25
|
We re-examined the final
proof from these slides
on Dedekind infinite sets.
This led to a discussion of different formulations
of the Axiom of Choice, in particular
that AC is equivalent to the statement
that for any set $A$ the set system
${\mathcal P}(A)\setminus\{\emptyset\}$
has a choice function.
We finished with a discussion of how to
modify the Principle of Induction
to the Principle of Strong Induction
and how to modify the Recursion Theorem
to the Course-of-Values Recursion Theorem.
|
Oct 27
|
We showed that the following two
potential definitions of `ordinal' are equivalent.
(a) an ordinal is a transitive set of transitive sets.
(b) an ordinal is a transitive set well-ordered by $\in$.
|
Oct 30
|
We proved Lemma 1 and half of Lemma 2:
Lemma 1.
-
An element of an ordinal is an ordinal.
-
The successor of an ordinal is an ordinal.
-
The union of a set of ordinals is an ordinal.
-
The intersection of a nonempty set of ordinals is an ordinal.
Lemma 2.
If $\alpha$ and $\beta$ are ordinals, then $\alpha\subseteq \beta$
and $\alpha\neq\beta$ if and only if $\alpha\in\beta$.
Quiz 7!
|
Nov 1
|
We completed the proof of
Lemma 2.
If $\alpha$ and $\beta$ are ordinals, then $\alpha\subseteq \beta$
and $\alpha\neq\beta$ if and only if $\alpha\in\beta$.
From Lemma 1 we derived
Corollary.
Every ordinal is the set of its predecessors.
($\alpha\in \textrm{ON}\Rightarrow \alpha
= \{\beta\in \textrm{ON}\;|\;\beta<\alpha\}$)
We then proved
Theorem.
The class
$\textrm{ON}$ of ordinal numbers
is a proper, transitive class that is well-ordered by $\in$.
|
Nov 3
|
We proved
Lemma.
If $\mathbb W$ is a well-ordered set and $f\colon \mathbb W\to \mathbb W$
is a morphism, then $f(x)\geq x$ for all $x$.
Corollary.
- No well-ordered set is isomorphic to a proper
initial segment of itself.
- The only automorphism of a well-ordered set is the identity function.
- If $\mathbb W_1$ and $\mathbb W_2$ are isomorphic well-ordered sets,
then the isomorphism from $\mathbb W_1$ to $\mathbb W_2$ is unique.
We stated
Theorem.
If $\mathbb W_1$ and $\mathbb W_2$ are well-ordered sets, then
either (i) $\mathbb W_1\cong \mathbb W_2$,
(ii) $\mathbb W_1$ is isomorphic to an initial segment of $\mathbb W_2$, or
(iii) $\mathbb W_2$ is isomorphic to an initial segment of $\mathbb W_1$.
|
Nov 6
|
We proved
Theorem.
If $\mathbb W_1$ and $\mathbb W_2$ are well-ordered sets, then
either (i) $\mathbb W_1\cong \mathbb W_2$,
(ii) $\mathbb W_1$ is isomorphic to an initial segment of $\mathbb W_2$, or
(iii) $\mathbb W_2$ is isomorphic to an initial segment of $\mathbb W_1$.
Quiz 8!
|
Nov 8
|
We proved
Theorem.
Any well-ordered set is isomorphic to an ordinal.
We discussed two versions of the Principle of Transfinite Induction.
|
Nov 10
|
We discussed:
(1) ON is well-ordered in both the set sense and the class sense.
(2) The well-orderedness of ON leads to a Principle of Transfinite
Induction
and various versions of Transfinite Recursion.
(We considered a basic version of Transfinite Recursion,
a parametrized version,
and an incomplete version.)
|
Nov 13
|
We discussed the order topology on an ordinal.
We explained how the recursive definitions
of sum and product of natural numbers may be extended
to the class of all ordinals.
We discussed visual interpretations of ordinal sum
and ordinal product.
Quiz 9!
|
Nov 15
|
We gave most of the proof of the associative law for ordinal
addition. What remains to show is that
\[
\bigcup_{\delta<\gamma} \alpha + (\beta+\delta) =
\bigcup_{\mu<\beta+\gamma} \alpha + \mu
\]
when $\gamma$ is a nonzero limit ordinal
and $x+(y+z) = (x+y)+z$
holds for every $z < \gamma$.
|
Nov 17
|
We completed the proof of the associative law for ordinal
addition. This involved introducing the concept
of cofinality.
We defined the Hartogs number of a set.
We showed that the Hartogs number of a set is an
initial ordinal (an aleph number).
We recalled the proof that every
set has a Hartogs number.
|
Nov 27
|
We showed that every set can be well-ordered.
We also explained why the transitive closure
of a set exists.
Quiz 10!
|
Nov 29
|
We defined the von Neumann Hierarchy
and explained why every element has a rank.
We then began discussing cardinal arithmetic,
starting with the ZF theorem that
AC $\Leftrightarrow$ for all infinite $B$,
$|B\times B|=|B|$. So far we have established:
Lemma. (ZF) Assume that $A$ is a set and $\langle H; < \rangle$
is a well-ordered set with $A$ disjoint from $H$.
If $|A\times H|\leq |A\cup H|$, then $|H|\leq |A|$ or $|A|\leq |H|$.
|
Dec 1
|
We heard two student presentations related to the Axiom of Choice.
The first was about the existence of a nonmeasurable set of real
numbers.
The second was about the Banach-Tarski paradox.
|
Dec 4
|
We finished the ZF-proof that “for all infinite $B$,
$(|B\times B|=|B|)$” implies the Axiom of Choice.
We sketched a ZF-proof that AC $\Leftrightarrow$ ZL.
Quiz 11!
|
Dec 6
|
We heard a student presentation comparing set theory
to type theory.
Review sheet for the Final Exam.
|
Dec 8
|
We completed the argument that any infinite
set is equipotent with its Cartesian square.
Review sheet for the Final Exam.
|
Dec 11
|
We discussed cardinal arithmetic, including:
Theorem. If $1\leq \kappa\leq \lambda$,
and $\lambda$ is infinite,
then $\kappa+\lambda=\kappa\cdot\lambda=\max(\kappa,\lambda)$.
Theorem. If $1\leq \kappa_{\alpha}$ for all $\alpha$,
and $\lambda$ is infinite, then
$\sum_{\alpha<\lambda} \kappa_{\alpha}= \sup(\kappa_{\alpha})\cdot\lambda = \max(\sup(\kappa_{\alpha}),\lambda)$.
Corollary. If $1\leq \kappa_0 < \kappa_1 < \cdots$
is a strictly increasing $\lambda$-sequence, then
$\sum_{\alpha<\lambda} \kappa_{\alpha}= \sup(\kappa_{\alpha})$.
Kőnig's Theorem. If $\kappa_{\alpha}<\lambda_{\alpha}$
holds for all $\alpha<\mu$, then
$\sum_{\alpha<\mu} \kappa_{\alpha} < \prod_{\alpha<\mu} \lambda_{\alpha}$.
Kőnig's Theorem implies Cantor's Theorem ($\kappa < 2^{\kappa}$).
If $\kappa$ is infinite, then $\kappa^{\kappa} = 2^{\kappa}$.
Kőnig's Corollary. If $\kappa$ is infnite,
then $\kappa < \kappa^{{\rm cf}(\kappa)}$.
Review sheet for the Final Exam.
|
Dec 13
|
We reviewed for the final following
this Review sheet.
[We decided to restrict final exam topics to exclude
Items (IX) (d)-(h) and practice problems (16) and (17)
of the review sheet.]
The Final Exam will be held at December 17, 7:30-10pm (Sunday),
in our usual classroom ECCR 108.
|