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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!