Lecture 1 (Sep 5)
We started thinking about some basic words like set and real number. These turn out to be quite tricky to define! Instead of defining what they are, we'll try to define them in terms of what properties they satisfy, with the hope that if we list sufficiently many properties, nothing else can satisfy all of them!
Click here for Tommy's lecture notesLecture 2 (Sep 9)
After discussing the syllabus in some detail, we returned to naive set theory, rewriting our expressions from last time using mathematical notation. Then we introduced two new operations with sets, set complement and cartesian product; the latter required us to invent a way to define ordered pairs in terms of sets. Finally, we gave an almost entirely rigorous definition of what a function is... except that we used the term "one" in our definition, which hasn't yet been defined. We'll patch this up next time!
Click here for Zoe's lecture notesLecture 3 (Sep 12)
We started by revisiting our formal definition of function from last time and making it completely analytic (ie rigorous and only dependent on previously defined concepts). We defined and played around with a bunch of associated concepts. One perhaps unexpected discovery was the preimages are nicer than images in that they play more nicely with set theory operations. Finally, we launched our first big project of the semester: listing a few properties that ℝ satisfies, with the hope that if we list enough such properties, only ℝ will satisfy them all!
Click here for Zoe's lecture notesLecture 4 (Sep 16)
We introduced 11 axioms for ℝ. As we went, we considered imposters: sets other than ℝ that satisfy all the axioms listed. By the end of class, we'd crossed off most of our imposters but still had a few left: ℚ, ℂ, the integers (mod 2). Despite this, we were able to prove some results directly from the axioms, eg the uniqueness of the additive and multiplicative identities and the uniqueness of additive and multiplicative inverses.
Click here for Zoe's lecture notesLecture 5 (Sep 19)
We started by proving that for any set satisfying (A1)-(A11), a×0=0 for any element a. (We also demonstrated that (A11) is necessary for this theorem to hold.) Next we added one more axiom: the order axiom (A12). This allows us to cross a few more imposters off our list: ℂ and the integers (mod p) aren't ordered. We used this to prove the important result that 1 > 0. Note that our proof isn't about ℝ; the result holds for any set satisfying (A1)-(A12).
Click here for Zoe's lecture notesLecture 6 (Sep 23)
We gave two meta-analytic proofs that the squareroot of 2 isn't rational. This gives us a way to distinguish between ℝ and ℚ. We codified this by introducing one last axiom, (A13), concerning the existence of the supremum of a set.
Click here for Zoe's lecture notesLecture 7 (Sep 26)
We proved the existence of an infimum of a set. Then we defined the positive integers, using the notion of successor sets. This gave an easy proof of induction, and we used induction to prove that 1 is the smallest positive integer. We also defined ℤ and ℚ formally.
Click here for Zoe's lecture notesLecture 8 (Sep 30)
We discussed strong induction, and used it to prove that the positive integers are well-ordered. We then formalized the notion of the floor of a number, and came up with a promising strategy for proving it. Along the way, we were led prove an important lemma: that there exist arbitrarily large positive integers.
Click here for Zoe's lecture notesLecture 9 (Oct 3)
We developed an outline of a proof of the existence and uniqueness of the floor and fractional part functions. (The details can be found in this write-up.) Next we proved a fundamental result: that √2 is a real number.
Click here for Zoe's lecture notesLecture 10 (Oct 7)
After a highly condensed review of our proof from last time that √2 is a real number, we switched gears and discovered that ℚ is dense in ℝ. (A clever trick deduced from this that the irrationals are also dense in ℝ.) We then started discussing (somewhat informally) how to compare sizes of infinite sets, formulating a meta-analytic definition of what it means for two sets to have the same size. This led us to the notion of countable, and we saw (meta-analytic) arguments that all of the set ℤpos+10, 2ℤpos, ℤ, and ℚpos are countable. Next time we'll make these notions more precise, and will encounter an uncountable set.
Click here for Zoe's lecture notesLecture 11 (Oct 17)
We proved that ℝ is uncountable and explored the sizes of a few other sets. This led us to formalize our notions by introducing the concept of bijection, injection, and surjection. We also briefly discussed the notorious Continuum Hypothesis.
Click here for Zoe's lecture notesLecture 12 (Oct 21)
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Lecture 13 (Oct 24)
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Lecture 14 (Oct 28)
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Lecture 15 (Oct 31)
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Lecture 16 (Nov 4)
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Lecture 17 (Nov 7)
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Lecture 18 (Nov 11)
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Lecture 19 (Nov 14)
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Lecture 20 (Nov 18)
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Lecture 21 (Nov 21)
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Lecture 22 (Nov 25)
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Lecture 23 (Dec 2)
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Lecture 24 (Dec 5)
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