Date

What we discussed/How we spent our time

Aug 26

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 28

We discussed the language of set theory.
We then discussed meaning of the axioms of Power Set,
Comprehension, Replacement, and Foundation.
We also discussed the Russell set and
the explained why naive set theory is inconsistent.

Aug 30

We discussed the meaning of the Axiom of Choice.
We discussed the fact that we may
form the intersection of any nonempty class of sets,
but we can only form the union of a set of sets.
We practiced writing the axioms formally.

Sep 4

We gave the Kuratowski definition of ordered pairs,
$(a,b)=\{\{a\},\{a,b\}\}$,
and proved the characteristic property of ordered pairs.
We defined ordered triples and $n$tuples.
We defined $A\times B$ and showed that it is a set
(if $A$ and $B$ are). We defined relations and functions.
We gave examples of different kinds of relations
in mathematics (equality, $\in$, order, betweenness,
adjacency, graphs of functions and operations).

Sep 6

We discussed terminology for functions.
We defined class functions.

Sep 9

We discussed the relationship between
coimages/partitions and kernels/equivalence relations.
Quiz 1!

Sep 11

We discussed strict and nonstrict orders.
Terminology: comparable/incomparable, maximum/minimum,
maximal/minimal, linear order/total order/chain,
antichain, wellfounded/wellordered.
We sketched a proof that wellordered = wellfounded+linearly ordered.

Sep 13

We discussed the principle of induction.
We worked on this handout.

Sep 16

We discussed the principle of strong induction.
We stated the Recursion Theorem.
Quiz 2!

Sep 18

We began to discuss the arithmetic
of $\mathbb N$, following
this handout.
Some hints!

Sep 20

We discussed more arithmetic
of $\mathbb N$ (distributive law, associative law for multiplication).
We discussed the meaning of ``Logic Is Backwards''.
We sketched the proof of the Recursion Theorem (pages 4749).

Sep 23

We discussed generalizations of the Recursion
Theorem: the parametrized version, the strong version,
and the version with partial functions.
Quiz 3!

Sep 25

We began to discuss ordinal and cardinal numbers.
Today
we discussed the meaning of
$A=B$
$A\leqB$
finite
infinite
countably infinite
countable
uncountable
We sketched the proof of the CantorBernsteinSchroeder Theorem.

Sep 27

We continued discussing ordinal and cardinal numbers,
and also mentioned the von Neumann hierarchy.
$n\in V_{n+1}$, $n\notin V_{n}$.
$V_{\omega}$ = hereditarily finite sets is a model of ZFC  Infinity.
$V_{\omega+\omega}$ = is a model of ZFC  Replacement.
Defined transitive sets, ordinals.
Stated the wellordering theorem.
defined cardinal numbers ( = initial ordinals).
Argued that equipotence classes of ordinals are bounded intervals.

Sep 30

We proved Cantor's Theorem.
We introduced $\aleph$ numbers.
Quiz 4!

Oct 2

We discussed why the sum rule and product rule
satisfy the recursion for $+$ and $\cdot$ on $\omega$.
Then we discussed why equipotence classes of finite
ordinals are singletons, why any subset of a finite
set is finite, and why $\omega$ is infinite.

Oct 4

We discussed:
$X\cup Y\leq X+Y$.
A finite union of finite sets is finite.
$X$ finite implies ${\mathcal P}(X)$ finite.
$X$ infinite implies $X>n$ for each $n\in\omega$ (without AC).
An infinite subset of a countably infinite set is countably infinite.
$X\leq \omega$ implies $X=\omega$ or $X=k$ for some $k\in\omega$.
A finite union of countable sets is countable (without AC).
Midterm Review!.

Oct 7

We discussed:
If $A$ is countably infinite and
$f:A\to B$ is surjective, then $B$ is finite
or $B=\omega$.
$\omega\times\omega=\omega$.
If $A=B=C=\omega$, then
$A=B\times C=\omega$.
If $A=\omega$, then $A=A^n$ for any $n\in\omega$.
A countable union of countable sets is countable.
(In ZFC, the statement stands as given. In ZF, the statement
stands provided we are given a sequence of
enumerations $\langle a_0, a_1,\ldots\rangle$
for the sequence $\langle A_0, A_1,\ldots\rangle$ of sets.)
Quiz 5!

Oct 9

Midterm Review!.

Oct 11

Midterm!.

Oct 14

We proved that the set of finite strings in a
nonempty countable alphabet is countably infinite.
We derived some corollaries, the most of important
of which are
${\mathcal P}_{\textrm{fin}}(\omega)=\omega$.
There are only countably many algebraic numbers.
No Quiz!

Oct 16

We defined addition, multiplication,
and exponentiation of cardinal numbers.
We discussed some rules of cardinal arithmetic.
We argued that $\mathbb R=2^{\aleph_0}$.
We argued that the number of continuous functions
$f:\mathbb R\to \mathbb R$ is $2^{\aleph_0}$, while
the number of all functions $g:\mathbb R\to \mathbb R$
is the strictly larger value
$\left \mathbb R^{\mathbb R}\right=2^{2^{\aleph_0}}$.

Oct 18

We argued that the following sets have size $2^{\aleph_0}$:
the set of irrational real numbers,
the set of infinite subsets of $\omega$,
the set of permutations of $\omega$.
We recalled the definition of ordinal number
and wellordered set. We proved that if
$f: \mathbb W\to \mathbb W$ is a
$<$homomorphism from a wellordered set
$\mathbb W$ to itself,
then it must be increasing in the sense that
$(\forall x)(f(x)\geq x)$ holds.

Oct 21

We discussed Theorem 6.1.3 and Corollary 6.1.5.
Quiz 6!
