Date
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What we discussed/How we spent our time
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Aug 24
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Syllabus. Text.
Some discussion about the foundations of mathematics.
The idea of a set. The language of set theory. The meaning of
"$2$", "$+$", "$=$" and "$4$", and proof that $2+2=4$.
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Aug 26
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The axioms. We discussed axioms 1, 3, 4, 5.
We wrote each of these axioms as a formal sentence.
We discussed the use of formulas
as abbreviations. We discussed a directed graph model of the axioms.
We defined the successor function $S(x) = x\cup \{x\}$,
and explained why the successor of a set is a set.
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Aug 28
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We discussed axioms 2, 6, 7. During the discussion
we described the Hilbert Hotel, Dedekind finiteness and Dedekind
infiniteness, and the notion of a class function. We defined these
words, phrases and symbols: inductive set, subset, $\subseteq$,
power set, ${\mathcal P}(X)$,
axiom schema. We noted the elementary theorem that $X=Y$ iff
$X\subseteq Y$ and $Y\subseteq X$.
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Aug 31
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We discussed Russell's paradox and impredicative definitions.
We showed that if $S\neq \emptyset$, then $\bigcap S$ is a set.
We discussed axiom 8.
Quiz 1.
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Sep 2
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We defined $(a,b)$ as $\{\{a\},\{a,b\}\}$, defined $A\times B$, then discussed
relations and functions.
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Sep 4
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We showed that $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$.
We discussed this handout.
We showed that every subset of a set $B$ is the image of a function
to $B$ and every partition of a set $A$ is the coimage of a function
from $A$.
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Sep 9
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We defined kernel of a function, equivalence relation,
and equivalence class. We
explained why every kernel of a function is an equivalence
relation. We sketched some ideas about how to prove that every
equivalence relation is the kernel of a function.
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Sep 11
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We proved that every equivalence relation is the kernel
of a function. We explained how to order
equivalence relations and partitions according
to coarseness/fineness.
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Sep 14
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This was a day of definitions and pictures, mostly. The definitions included:
strict and nonstrict partial orders, the antisymmetry,
irreflexivity and asymmetry properties of a binary relation.
We pointed out that a binary relation is asymmetric iff it is irreflexive
and antisymmetric, and that to each strict partial order there is an
associated partial order, i.e.:
$$
\leq\; =\; (\lt\cup =)\quad\textrm{and}\quad
\lt \;=\; (\leq - =).
$$
We gave the meanings of
and the words and phrases:
poset, total order, comparable/incomparable, chain/antichain,
maximal/minimal, maximum/minimum, cover, duality/dual, supremum/lub/join,
infimum/glb/meet.
Quiz 2.
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Sep 16
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We defined the natural numbers, $\omega$.
We showed that $\omega$ is an inductive set contained in every
inductive set.
We proved the principle of induction.
We defined $m\lt n$ and $m\leq n$. We proved that
$0\leq n$ for all $n$, that $m\lt S(n)\leftrightarrow m\leq n$,
and that $\lt$ is transitive on $\omega$.
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Sep 18
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We completed the proof that $\omega$
is strictly linearly ordered by $\in$.
We showed that $m\lt n\leftrightarrow m\subsetneq n$,
We proved the principle of strong induction.
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Sep 21
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We defined well-ordered and
well-founded sets.
We observed that well-ordered = well-founded + linearly ordered.
We proved that induction is valid over an ordered set iff the ordered set
is well-founded, and derived from this that $\omega$
is well-ordered. We then turned to the recursion theorem,
stating the Recursion Theorem and stating the recursive definitions of the
arithmetic operations on $\omega$.
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Sep 23
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We proved some
laws of arithmetic.
Quiz 3.
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Sep 25
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We proved the Recursion Theorem.
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Sep 28
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We discussed recursion with parameters and course of values
recursion.
Quiz 4.
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Sep 30
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We defined $|A|=|B|$, $|A|\leq |B|$, finite, infinite.
We proved that equipotence is a class equivalence relation.
We proved the Cantor-Bernstein Theorem.
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Oct 2
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We discussed properties of finite sets. (Section 4.2).
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Oct 5
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We introduced the Axiom of Foundation, completing
our introduction of the axioms of ZF. We discussed
the distinction between finiteness and Dedekind finiteness,
noting that the former implies the latter
and that if $A$ is Dedekind finite but not finite,
then $n\lt |A|$ for every $n$ while $|\omega|\not\leq |A|$.
We defined amorphous set.
Quiz 5.
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Oct 7
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We continued our discussion of infinite Dedekind finite sets
and the existence of amorphous sets.
We talked about the problem of showing
that $\Sigma\to \sigma$ when
$\Sigma\cup\{\sigma\}$ is a set of first-order sentences,
especially when $\Sigma = ZF$.
We stated Gödel's Completeness Theorem and his
Second Incompleteness Theorem, and explained their connection
to this problem. We mentioned three
techniques for modifying models of $ZF$, namely
Inner Models, Extensions (forcing), and
embedding models of $ZFA$ into models of $ZF$
via the Jech-Sochor Embedding Theorem
(also forcing).
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Oct 9
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We began discussing countably infinite sets.
We showed that an infinite set is Dedekind finite
iff it has no countably inifnite subset.
We showed that a subset of a countably infinite set
is finite or countably infinite. We showed that the image
of a countable set under a function is at most
countable. We discussed a theorem that says, roughly,
if it is possible to describe the elements of a set $B$ in a language
based on a finite or countable alphabet, then $B$ is at
most countable. We used this to show that the sets:
$\mathbb Z$, $\mathbb Q$, $\overline{\mathbb Q}$, and
the set of formulas of set theory are each countable.
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Oct 12
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We proved that if $A$ and $B$ are countably infinite, then so are
$A\dot{\cup}B$,
$A{\cup}B$,
$A\times B$, and
$A^*$. We proved Cantor's Theorem that the set of real numbers
is uncountable.
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Oct 14
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We discussed the review sheet.
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Oct 16
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Midterm.
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Oct 19
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We defined characteristic functions and
proved that $|\mathbb R| = |(a,b)| = |2^{\omega}|$
and $|2^A|=|\mathcal P(A)| > |A|$. We then
started discussing ordinal numbers.
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Oct 21
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We discussed well-ordered sets.
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Oct 23
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We discussed transitive sets and ordinals.
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Oct 26
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We talked about sets versus classes, and proved that ON (the class of ordinal
numbers) is a proper class that is transitive and well-ordered by $\in$.
We defined limit ordinal, and mentioned that if $X$ is a set of
ordinals, then $\bigcup X$ is the least upper bound of $X$.
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Oct 28
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We discussed transfinite induction and recursion.
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Oct 30
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We discussed ordinal arithmetic, including visual interpretations of
ordinal addition and product. To introduce language to
describe the visual images we defined
the ordinal sum $P\oplus Q$
of arbitrary posets, and the lexicographic and antilex
ordering on a product of posets. We started a proof of the associative
law of ordinal addition, but our proof used an unproved lemma:
$\textrm{sup}\{\alpha+(\beta+\mu) \mid \mu<\gamma\}=
\alpha+\textrm{sup}\{\beta+\mu \mid \mu<\gamma\}$ for $\gamma$ limit.
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Nov 2
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We discussed the leftover claim,
$\textrm{sup}\{\alpha+(\beta+\mu) \mid \mu<\gamma\}=
\alpha+\textrm{sup}\{\beta+\mu \mid \mu<\gamma\}$ for $\gamma$ limit,
from last Friday's lecture. We also mentioned Cantor Normal Form
and $\varepsilon_0$.
Quiz 6.
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Nov 4
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(Read 129-133.)
We defined initial ordinals and enumerated them by the
$\omega$-function: $\omega_0 = \omega$, $\omega_{\alpha}$ =
least inital ordinal strictly larger than $\omega_{\beta}$ for
all $\beta<\alpha$. We noted that every well-orderable set is
equipotent with a unique initial ordinal, which is its cardinal number.
We proved Hartogs Theorem and
defined the Hartogs number, $h(X)$, of a set $X$. We introduced the
notation $\aleph_{\alpha}$ where $\alpha$ is an ordinal.
We defined the operations of cardinal addition, multiplication
and exponentiation.
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Nov 6
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We showed that $|A|\leq |A\times A|$ for any set $A$, that
$|A|\lt |A\times A|$ for any amorphous set $A$, and that
$|A|=|A\times A|$ for any well-orderable set $A$.
We showed that the same (in)equalities hold with
disjoint union in place of cartesian product.
We also argued that if $\kappa$ and $\lambda$ are cardinals, then
$\max\{\kappa,\lambda\}$=$\kappa+\lambda$=$\kappa\cdot\lambda$ if
$\lambda\gt 0$ and $\kappa$ is infinite.
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Nov 9
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We introduced the Axiom of Choice and noted some
immediate consequences.
Quiz 7.
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Nov 11
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We discussed the question: how do we know we can choose one
element from a nonempty set? We concluded that it is
possible, and by induction that in ZF there is a choice
function
for any finite set of nonempty sets. We then proved some
slight modifications of the axiom of choice are equivalent to the axiom.
Then we proved that AC is equivalent to WO (the Well-Ordering Principle).
We introduced the statement of Zorn's Lemma.
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Nov 13
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We proved that Zorn's Lemma is equivalent to the Axiom of Choice.
We proved that every vector space has a basis.
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Nov 16
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We compared two proofs of the fact that every vector space has a basis.
Quiz 8.
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Nov 18
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We proved that if $|A\times A| = |A|$ for all infinite $A$,
then every set can be well-ordered. Then we proved that
it is possible to partition $\mathbb R^3$ into disjoint unit circles
in such a way that no two circles lie in the same plane.
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Nov 20
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(Read 129-133.)
Cardinal arithmetic. We defined infinite sums and products of cardinals,
and cofinality, and showed:
if $(\kappa_i)_{i<\lambda}$
is an infinite sequence of nonzero cardinals, then
$\sum_{i<\lambda} \kappa_i = \lambda\cdot \sup\{\kappa_i\}$.
if $(\kappa_i)_{i<\lambda}$
is an infinite sequence of nonzero cardinals, then
$\prod_{i<\lambda} \kappa_i = \sup\{\kappa_i\}^{\lambda}$.
(König's Theorem) If $\kappa_i<\lambda_i$ for all $i$,
then $\sum \kappa_i<\prod \lambda_i$.
(König's Corollary) $\kappa<\kappa^{\textrm{cf}(\kappa)}$.
In the process of establishing the first two we had to prove an interesting
claim: If $\lambda$ is an infinite cardinal, then $\lambda$ can be
partitioned into $\lambda$-many cofinal subsequences. This argument used
the canonical well-order on $\lambda\times\lambda$.
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Nov 30
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We reviewed the Nov 20 lecture, including the definition
of cofinality. We observed that for any infinite cardinal
$\kappa$ there is a field $\mathbb F$ of cardinality
$\kappa$, and that if, for some infinite cardinal $\lambda$, $V$ is
a $\lambda$-dimensional $\mathbb F$-vector space, then
$|V| = \kappa\cdot\lambda$ and $|V^*| = \kappa^\lambda$.
We used this to motivate the Exponentiation Problem:
Given $\alpha$ and $\beta$, determine $\gamma$ so that
$\aleph_{\alpha}^{\aleph_{\beta}}=\aleph_{\gamma}$.
We introduced the $\aleph$-function,
the $\beth$-function and the $\gimel$-function.
We discussed CH and GCH.
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Dec 2
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We discussed regular versus singular cardinals,
and successor versus limit cardinals. We proved
that successor cardinals are regular,
and noted that $\aleph_{\alpha+\omega}$ is singular
for every $\alpha$. We showed that $\textrm{cf}(2^{\kappa})>\kappa$,
and derived that $|\mathbb R|\neq \aleph_{\omega}$, in fact
$|\mathbb R|$ has uncountable cofinality. We discussed
the obvious properties of the continuum function, and then
proved the nonobvious property that
$2^{\lambda}=(2^{<\lambda})^{\textrm{cf}(2^{\lambda})}$
whenever $\lambda$ is limit. From this we derived that if
$\lambda$ is singular and the continuum function
is constant $\kappa$ below $\lambda$, then $2^{\lambda}=\kappa$.
I circulated a review sheet.
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Dec 4
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We reviewed what has been proved about the continuum function,
and stated Easton's Theorem.
We then began discussing what is known about
general exponentiation, including
Shelah's Theorem that $\gimel(\aleph_{\omega})=(\aleph_{\omega})^{\omega}$
is either equal to $2^{\omega}$ or is
$\lt\aleph_{\omega_4}$. We then began the proof of the
Cardinal Exponentiation Theorem. So far we have proved that
(i) if $\kappa\leq\lambda$, then $\kappa^{\lambda}=2^{\lambda}$,
and (ii) if there exists $\mu\lt\kappa$ such that
$\kappa\leq\mu^{\lambda}$, then $\kappa^{\lambda}=\mu^{\lambda}$.
What remains to consider is the case where $\lambda<\kappa$
and there is no $\mu<\kappa$ such that $\kappa\leq \mu^{\lambda}$.
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Dec 7
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We proved Hausdorff's Formula,
$\aleph_{\alpha+1}^{\aleph_{\beta}}=
\aleph_{\alpha}^{\aleph_{\beta}}\cdot\aleph_{\alpha+1}$.
Then we proved a lemma that says
if $\kappa$ is a limit cardinal
and $\textrm{cf}(\kappa)\leq \lambda$,
then
$\kappa^{\lambda}=
(\sup_{\alpha\to\kappa}(|\alpha|^{\lambda}))^{\textrm{cf}(\kappa)}$.
Quiz 9.
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Dec 9
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We completed the proof of the Cardinal Exponentiation Theorem.
We observed that $\kappa^{\lambda}$ is either
$2^{\lambda}, \kappa^{\textrm{cf}(\kappa)}, \kappa$, or
$\mu^{\textrm{cf}(\mu)}$ for $\mu$ the least cardinal $\lt\kappa$
such that $\mu^{\lambda}\geq\kappa$. We derived a simpler
Cardinal Exponentiation Theorem under GCH. We also
showed that GCH is equivalent to the statement that
$\kappa^+=\gimel(\kappa)$ for all infinite $\kappa$.
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