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 Date What we discussed/How we spent our time Aug 26 Syllabus. Policies. Text. Review of Math 2001. We discussed how the axioms of set theory allow the construction of the natural numbers, $\mathbb N = \langle \{0,1,2,\ldots\}; 0, 1, +, \cdot\rangle$. Aug 28 We discussed how to construct $\mathbb Z$ from $\mathbb N$, $\mathbb Q$ from $\mathbb Z$, and we started discussing how to construct $\mathbb R$ from $\mathbb Q$. Aug 30 We defined finite sequences, infinite sequences, the limit of a sequence of rational numbers, Cauchy sequences of rational numbers, null sequences. We defined a real number to be an equivalence class of Cauchy sequences of rational numbers modulo null sequences. We explained how to define $0, 1, +, -, \cdot, <$ on $\mathbb R$. This handout on ordered fields was circulated. Sep 4 We discussed the axioms for ordered fields, and gave some examples and nonexamples. Sep 6 We worked on this handout. Then we discussed why any ordered field contains an isomorphic copy of $\mathbb N$. Sep 9 We introduced maximum elements, upper bounds, least upper bounds, suprema, and the completeness axiom. [The dual concepts are minimum elements, lower bounds, greatest lower bounds, infima, and the completeness axiom.] Quiz 1. Sep 11 We discussed a construction of a nonarchimedean ordered field, $\mathbb R(t)$, whose elements are rational functions in the variable $t$ with coefficients in $\mathbb R$. The order is defined by saying that $\frac{a_mt^m+\cdots+a_1t+a_0}{b_nt^n+\cdots+b_1t+b_0}$ is positive in $\mathbb R(t)$ iff $\frac{a_m}{b_n}$ is positive in $\mathbb R$. We will see that the completeness property does not hold in $\mathbb R(t)$. We started investigating consequences of the Completeness Axiom which hold in any complete ordered field, but fail in $\mathbb R(t)$. Today we showed that the Nested Interval Property holds in any complete ordered field. Sep 13 We showed that a complete ordered field $\mathbb F$ has the following properties: $\mathbb F$ is archimedean. $\mathbb Q_{\mathbb F}$ is dense in $\mathbb F$. $\mathbb F = \bigcup_{n=1}^{\infty} [-n,n]$. For every $r\in \mathbb F$ there exists $m\in\mathbb Z_{\mathbb F}$ such that $m-1\leq r< m$. We examined $\mathbb F = \mathbb Q$ and $\mathbb R(t)$ to see which of these examples has the Nested Interval Property, the Archimedean Property, or has a dense set of rationals. Sep 16 We discussed this handout on cardinality. Quiz 2. Sep 18 We continued to discuss cardinality. We proved the Cantor-Bernstein-Schroeder Theorem, and explained why $|\mathbb N| = |\mathbb Z| = |\mathbb Q|$. Sep 20 We discussed Cantor diagonalization''. We used it to prove (i) $[0_{\mathbb R},1_{\mathbb R}]$ is uncountable, (ii) ${\mathcal P}(\mathbb N)$ is uncountable, and (iii) there is no surjective function from a set $X$ to ${\mathcal P}(X)$ for any set $X$. We explained why the following three sets have the same size: (i) ${\mathcal P}(\mathbb N)$, (ii) the set of (characteristic) functions $f:\mathbb N\to \{0,1\}$, and (iii) the set of paths through a complete binary tree. We proved that if $\mathbb F$ is an ordered field satisfying the Nested Interval Property, then the cardinality of its unit interval $[0_{\mathbb F}, 1_{\mathbb F}]$ is at least $|{\mathcal P}(\mathbb N)|$. (We did this by embedding the set of paths through the complete binary tree into the set $[0_{\mathbb F}, 1_{\mathbb F}]$.) We stated that if $\mathbb F$ is an ordered field satisfying the Archimedean Property, then the cardinality of its unit interval $(0_{\mathbb F}, 1_{\mathbb F})$ is at most $|{\mathcal P}(\mathbb N)|$. (Proof postponed until next lecture.) Sep 23 We discussed the following results. An ordered field is Archimedean iff its rationals are dense. If $\mathbb F$ is an Archimedean ordered field and $\mathbb K$ is a complete ordered field, then there is a unique embedding of $\mathbb F$ into $\mathbb K$. Any two complete ordered fields are isomorphic. An ordered field is complete iff it is Archimedean and satisfies the Nested Interval Property. The cardinality of an ordered field with the Nested Interval Property is at least $|{\mathcal P}(\mathbb N)|$; The cardinality of an Archimedean ordered field is at most $|{\mathcal P}(\mathbb N)|$; the cardinality of a complete ordered field is exactly $|{\mathcal P}(\mathbb N)|$. Quiz 3. Sep 25 We discussed limits of sequences. Most of the discussion was about how to read, write, and understand formal sentences, like $$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (|a_i-L|<\varepsilon)).$$ The main concepts/terms discussed were: The alphabet of symbols appropriate for writing about a structure (variables, logical symbols, nonlogical symbols, punctuation). Defined symbols (we gave the formula for $y=|x|$'' that is valid for any ordered field). Sentence formation rules. Formula trees. Restricted quantifiers. Truth versus provability. Quantifier games for determining truth (with $\forall$belard and $\exists$loise). In particular, we discussed how to describe a winning strategy for $\exists$ or $\forall$. Logic. Practice! Sep 27 We defined a metric, or distance function. (Defining properties: a metric on a set $A$ is a 2-variable function $d: A\times A\to \mathbb R$ that is positive definite, symmetric, and satisfies the triangle inequality.) The metric used in $\mathbb R$ is $d(a,b) = |a-b|$. So the definition of limit, $$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (|a_i-L|<\varepsilon)),$$ should be thought of in the following form: $$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (d(a_i,L)<\varepsilon)).$$ We gave a fake proof that limits are unique. Then we gave a correct proof. Finally, we proved that a convergent sequence is bounded. Sep 30 We discussed the Algebraic Limit Theorem. Quiz 4. Oct 2 We discussed the Order Limit Theorem and the Monotone Convergence Theorem. Midterm Review!. Oct 4 Practice with the Monotone Convergence Theorem! Oct 7 We defined subsequences, and discussed the theorem that if $(a_n)_{n\in\mathbb N}$ converges to $L$, then any subsequence also converges to $L$. From this we developed a divergence criterion. Quiz 5. Oct 9 Midterm Review!. Oct 11 Midterm! Oct 14 We reviewed the meaning and importance of definitions'' and winning strategies''. We proved the Bolzano-Weierstrass Theorem. No Quiz! Oct 16 We recalled the definition of Cauchy sequence. We worked on a handout to prove that any convergent sequence is bounded. We began a discussion of why any Cauchy sequence of real numbers is convergent. Oct 18 We finished the proof of the Cauchy Criterion. We discussed different kinds of limiting processes, in particular infinite series. Oct 21 We discussed the definition of $\sum a_i = L$'', and the following facts about series. Cauchy Criterion for series. Comparison Test. Series of positive terms converge iff their partial sums are bounded. $N$th term test for divergence. Deleting first $N$ terms does not affect convergence. The Harmonic Series diverges. The sum of the even terms of the Harmonic series divereges, as does the sum of the odd terms. Quiz 6.