Nadja, on Gaara from Naruto: "It would be great to take him to the beach. We'd have the most kick-ass sandcastle ever."
So Friday we had Mr. Sally's little additional placement test, and we got chapter 1 of his book[3]. He told us to do all the exercises from it for Monday, "but for God's sake, don't write them up!" Of course, I missed that last bit. Chapter 1 is very basic set theory and algebra, mostly very tedious. I wasted nearly all of Saturday writing it, until finally today Lucas told me we weren't supposed to do that, and reminded me that we *were* supposed to write up the test problems we hadn't gotten. Yay.
There is one thing in Chapter 1 I didn't know how to do, though. At the end of it, he proves the equivalence of Axiom of Choice, Zorn's Lemma, Hausdorff Maximality Principle, and Well-Ordering Principle. Mostly. He does it by the circle AC⇒weakened form of ZL⇒Hausdorff⇒ZL⇒WOP⇒AC, but he leaves the last 2 steps as exercises. Well, WOP⇒AC is trivial, but deriving well-ordering from Zorn's Lemma? Every poset I could think of either didn't meet the conditions or wouldn't have a well-ordering of the set as a maximal element. So actually on Saturday I often alternated between doing nothing because the tedious problems bored me, and doing nothing because I was getting nowhere on ZL⇒WOP. Today I eventually ended up asking Lucas how to do it, and that was definitely a "D'oh!" solution. Oh well.
Now, one thing you may notice there is how he proves Zorn's Lemma. He doesn't do it directly from AC by that awful transfinite-recursion-if-there's-no-maximal-element-we-could-fit-all-the-ordinals-in-the-set argument. Instead, he first proves, with no AC, that if you have a poset X where every chain has a supremum, and a function f:X→X st f(x)≥X ∀x, then f has a fixed point. Then, if you have a poset where every chain has a supremum, there must be a maximal element as otherwise you could define a function st f(x)>x ∀x by f(x)=g({y∈X:y>x}), where g is a choice function for X. Of course, this is not quite Zorn's Lemma, but as so often Zorn's Lemma is used where the partial order is set inclusion and the upper bound is the union of the chain, it's often enough in practice... and specifically, it's enough to prove Hausdorff Maximality Principle, which obviously implies the full Zorn's Lemma.
Of course, I haven't actually stated the proof of that fixed-point lemma, so if anyone wants to see it, here it is.
( Much math )
-Sniffnoy, who still must do his laundry (actually I was supposed to do that yesterday), and finish problem 3 from the test
[3]According to the syllabus, this book is currently titled Kick-ass Mathematics.
[4]This proof was not exactly directly copied from the one in the book, but the "Whew" at the end is.
So Friday we had Mr. Sally's little additional placement test, and we got chapter 1 of his book[3]. He told us to do all the exercises from it for Monday, "but for God's sake, don't write them up!" Of course, I missed that last bit. Chapter 1 is very basic set theory and algebra, mostly very tedious. I wasted nearly all of Saturday writing it, until finally today Lucas told me we weren't supposed to do that, and reminded me that we *were* supposed to write up the test problems we hadn't gotten. Yay.
There is one thing in Chapter 1 I didn't know how to do, though. At the end of it, he proves the equivalence of Axiom of Choice, Zorn's Lemma, Hausdorff Maximality Principle, and Well-Ordering Principle. Mostly. He does it by the circle AC⇒weakened form of ZL⇒Hausdorff⇒ZL⇒WOP⇒AC, but he leaves the last 2 steps as exercises. Well, WOP⇒AC is trivial, but deriving well-ordering from Zorn's Lemma? Every poset I could think of either didn't meet the conditions or wouldn't have a well-ordering of the set as a maximal element. So actually on Saturday I often alternated between doing nothing because the tedious problems bored me, and doing nothing because I was getting nowhere on ZL⇒WOP. Today I eventually ended up asking Lucas how to do it, and that was definitely a "D'oh!" solution. Oh well.
Now, one thing you may notice there is how he proves Zorn's Lemma. He doesn't do it directly from AC by that awful transfinite-recursion-if-there's-no-maximal-element-we-could-fit-all-the-ordinals-in-the-set argument. Instead, he first proves, with no AC, that if you have a poset X where every chain has a supremum, and a function f:X→X st f(x)≥X ∀x, then f has a fixed point. Then, if you have a poset where every chain has a supremum, there must be a maximal element as otherwise you could define a function st f(x)>x ∀x by f(x)=g({y∈X:y>x}), where g is a choice function for X. Of course, this is not quite Zorn's Lemma, but as so often Zorn's Lemma is used where the partial order is set inclusion and the upper bound is the union of the chain, it's often enough in practice... and specifically, it's enough to prove Hausdorff Maximality Principle, which obviously implies the full Zorn's Lemma.
Of course, I haven't actually stated the proof of that fixed-point lemma, so if anyone wants to see it, here it is.
( Much math )
-Sniffnoy, who still must do his laundry (actually I was supposed to do that yesterday), and finish problem 3 from the test
[3]According to the syllabus, this book is currently titled Kick-ass Mathematics.
[4]This proof was not exactly directly copied from the one in the book, but the "Whew" at the end is.
