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## Lecture Topics

 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, well-founded/well-ordered. We sketched a proof that well-ordered = well-founded+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 47-49). 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 $|A|=|B|$ $|A|\leq|B|$ finite infinite countably infinite countable uncountable We sketched the proof of the Cantor-Bernstein-Schroeder 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 well-ordering 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 well-ordered set. We proved that if $f: \mathbb W\to \mathbb W$ is a $<$-homomorphism from a well-ordered 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! Oct 23 We worked through almost all of Section 2 of Chapter 6. Our ultimate goal was to prove that the set ON of ordinal numbers is a proper class that is well ordered (in the class sense) by $\in$. Oct 25 We proved that ON is a proper class that is well ordered (in the class sense) by $\in$. We discussed Russell's Paradox, Cantor's Paradox, and the Burali-Forti Paradox. Oct 28 We discussed transfinite induction and recursion. We defined ordinal arithmetic. We defined what it means for a function defined on the ordinals to be continuous. Quiz 7! Oct 30 We discussed how to visualize the arithmetic operations on the ordinals. We also defined the order type'' of a well-ordered set, and the lexicographic'' and antilexicographic'' orderings of products and sets of strings. Nov 1 We discussed a transfinite induction proof. We briefly mentioned the idea behind Cantor Normal Form and the definition of $\varepsilon_0$. Finally we discussed the definition of alephs. Nov 4 We discussed $\aleph$ numbers and the possible meaning of cardinality'' in the absence of the axiom of choice. We got off onto a side discussion of the question: Is it true that $|\omega|\leq |A|$ when $A$ is infinite'' in the absence of choice? Quiz 8! Nov 6 We reviewed the quiz. We defined the Hartogs number of a set. We showed that the Hartogs number of a set, if it exists, is an initial ordinal (an aleph number). Using the Axiom of Replacement, we showed that every set has a Hartogs number. We finished the lecture by proving the following technical lemma, which will be used to prove Tarski's Theorem: Lemma. (ZF) If $H$ is a well-orderable set that is disjoint from some set $A$, and $|A\times H| = |A\cup H|$, then either $|A|\leq |H|$ or $|H|\leq |A|$. Nov 8 We proved (in ZF) that if every infinite set satisfies $|B\times B|=|B|$, then every infinite set is well-orderable. We began a proof of the converse, but did not finish. Nov 11 We finished the proof that if every set is well-orderable, then any infinite set $B$ must satisfy $|B\times B|=|B|$. We remarked that an ordinal $\alpha$ has the property that the canonical well-ordering of $\alpha\times\alpha$ has order type $\alpha$ iff $\alpha = 0, 1$ or $\alpha = \omega^{\omega^\beta}$ for some ordinal $\beta$. We derived from the main theorem that if $\lambda$ and $\kappa$ are finite cardinals or alephs and at least one is infinite, then $\lambda+\kappa = \lambda\cdot\kappa = \max(\lambda,\kappa)$. Quiz 9! Nov 13 We discussed this handout on the Axiom of Choice. We explained the equivalence of (1), (2), (3), (4), (5), and we proved that (14) implies (1). Nov 15 We worked on the proof that AoC $\equiv$ Well-Ordering Theorem $\equiv$ Zorn's Lemma. Nov 18 We finished the proof that AoC $\equiv$ Well-Ordering Theorem $\equiv$ Zorn's Lemma. Quiz 10! Nov 20 We showd the AC is equivalent to the statement that any two sets have comparable cardinality. We sketched two proofs that every vector space has a basis, one using the Well-Ordering Theorem and the other using Zorn's Lemma. Nov 22 We started discussing cardinal arithmetic by defining infinite sum and product. We also mentioned the relation between $\leq$ and $\leq_*$. We showed that $\sum_{\alpha<\lambda} \kappa_{\alpha} = \kappa\cdot\lambda$ if $\kappa_{\alpha}>0$ for all $\alpha$ and if $\kappa := \sup(\kappa_{\alpha})$. We proved König's Theorem, from which we derived Cantor's Theorem (and claimed that $\kappa<{\kappa}^{\textrm{cf}(\kappa)}$). Dec 2 We defined cofinality, and proved some basic things about the cofinality of ordinals and cardinals, such as $\textrm{cf}(\alpha)\leq \alpha$. $\textrm{cf}(0) = 0$, hence $0$ is regular. Every successor ordinal has cofinality $1$, hence $1$ is the only regular successor ordinal. Every infinite limit ordinal has cofinality $\geq \omega$. In particular, $\omega$ is regular. $\textrm{cf}(\aleph_{\omega}) = \omega$, so $\aleph_{\omega}$ is singular. $\textrm{cf}(\textrm{cf}(\alpha))=\textrm{cf}(\alpha)$. So, if $\alpha$ is infinite, then $\textrm{cf}(\alpha)$ is an infinite, regular cardinal. Any cofinal subset of an ordinal $\alpha$ contains a cofinal subset of cardinality $\textrm{cf}(\alpha)$. An infinite cardinal $\kappa$ is a (disjoint) union of $\lambda$-many subseteq of size $<\kappa$ iff $\textrm{cf}(\kappa)\leq\lambda$ Notes on Cardinal Arithmetic! No Quiz! Dec 4 We defined the continuum function $C(\kappa) = 2^{\kappa}$, the aleph function $\aleph_{\alpha}$, the beth function $\beth_{\alpha}$, and the gimel function $\gimel(\kappa) = \kappa^{\textrm{cf}(\kappa)}$. We stated CH ($\aleph_1=\beth_1$) and GCH ($\aleph_{\alpha}=\beth_{\alpha}$ for all $\alpha$) in terms of the functions $\aleph$ and $\beth$. We reviewed some elements of the Cardinal Arithmetic handout. We explained why Theorems 2, 3, 4 and Corollary 5 are true. Dec 6 Samuel recited the following poem he wrote! There was a time, when we were young, we thought infinity was never wider than the sky, nor deeper than the sea. But now we've gazed into the depths of "omega" and beyond; and sit, in dumbstruck silence, now, our innocence all gone. So let us not attempt to see what must remain unseen; let us rejoice in finite things, in one, and two, and three: For if we crack and crush our minds with things too huge to be known, we'll end up deranged, and lost in our brains, as Cantor and Gödel have shown. So have no fear, O students, professors: infinity cares for herself. Our work is to care for the simpler things; let us do so, that we may be well. Then we proved that successor cardinals are regular. We established that $\textrm{cf}\left(2^{\kappa}\right)>\kappa$. We discussed Easton's Theorem. We proved the first two cases of the Main Theorem of Cardinal Arithmetic. (These are the cases where $\lambda$ is large relative to $\kappa$ or very small relative to $\kappa$.) Review Sheet for Final!. Dec 9 No Quiz! We finished the proof of the Main Theorem of Cardinal Arithmetic. We pointed out that if $V$ is a $\lambda$-dimensional vector space (where $\lambda$ is infinite) over a field $\mathbb F$ of cardinality $\kappa$, then $|V^*| = \dim(V^*) = \kappa^{\lambda}$. Here $V^*$ is the dual space of $V$. (This is a real-life'' situation where one might need to compute $\kappa^{\lambda}$.) Dec 11 We reviewed for the final exam.