170- Midterm 3

November 21, 2010

The third midterm is here.

Solutions follow.

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“The shadow scholar”

November 16, 2010

A must read (but depressing) article in The Chronicle of Higher Education: The Shadow Scholar. The man who writes your students’ papers tells his story. By “Ed Dante”.

Two examples:

From my experience, three demographic groups seek out my services: the English-as-second-language student; the hopelessly deficient student; and the lazy rich kid.

For the last, colleges are a perfect launching ground—they are built to reward the rich and to forgive them their laziness. Let’s be honest: The successful among us are not always the best and the brightest, and certainly not the most ethical. My favorite customers are those with an unlimited supply of money and no shortage of instructions on how they would like to see their work executed. While the deficient student will generally not know how to ask for what he wants until he doesn’t get it, the lazy rich student will know exactly what he wants. He is poised for a life of paying others and telling them what to do. Indeed, he is acquiring all the skills he needs to stay on top.

And:

The 75-page paper on business ethics ultimately expanded into a 160-page graduate thesis, every word of which was written by me.

Edit: The article has been commented at the always interesting Making Light. Here is the link, read the comments.


170- Quiz 10

November 16, 2010

Quiz 10 is here.

Solutions follow.

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170- Quiz 9

November 11, 2010

Quiz 9 is here.

Solutions follow.

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170- Extra credit problem

November 9, 2010

This problem is due Tuesday November 30, and will replace your lowest quiz score, if you choose to turn it in. Additional extra credit is possible depending on the quality of your work.

Please work on your own. If somebody helps you, or you find ideas or the solution either in a book or online, please mention this in what you turn in, including the name of the person, the name of the book, or a link to the relevant website, together with any additional information that may be useful to identify the sources.

In lecture we explained how Newton’s method can be used to approximate \sqrt2 numerically: We start with a guess x_0, and then define

\displaystyle x_1=x_0-\frac{x_0^2-2}{2x_0},

\displaystyle x_2=x_1-\frac{x_1^2-2}{2x_1},

\displaystyle x_3=x_2-\frac{x_2^2-2}{2x_2},

etc.

The goal here is to see how good these approximations are.

  1. Show that if x_0>0, then x_1,x_2,\dots are all larger than 1.
  2. Show that if x_0\ge1, then {}|x_1^2-2|<|x_0^2-2|^2/4, {}|x_2^2-2|<|x_1^2-2|^2/4, etc.
  3. If we want to compute \sqrt2 with an accuracy of 12 significant digits, and we begin with x_0=1, how many iterations do we need to perform?

The perils of grading

November 7, 2010

There is this cemetery near our house and, near the cemetery, there is a playground Francisco wanted to visit.

Off we go, crossing the cemetery. Najuma and Francisco are ahead of me, he busy picking up pine cones, she pulling the wagon that Francisco insisted we take with us. I am distracted, reading the last few pages of a homework set I want to finish grading.

There is a wooden sort of plank on the ground. You see where this is going, don’t you?

Anyway, I did not even get to step on the thing, wafer-thin as it was. As I am putting my foot down, I realize the mistake. Gravity is a marvelous thing; it is the first thing I tried to teach Francisco, 3 years or so ago, how gravity works; this is just a practical lesson. Because the wood breaks and falls and I fall with it, 6 feet or so. Najuma said later that she heard a loud noise, a rumbling coming from the ground. She turned, and saw me disappear.

The two of them approach as I try to get up, and somehow manage to find enough strength to lift myself out of the grave. (As a kid, this is the sort of thing I never had enough strength to do. The one time I went rock climbing, it was what stopped me from reaching higher than I did.) Francisco thinks it is a game and wants to go down the hole as well. Crying ensues.

And that’s it, really. Somehow I managed not to injure myself. Except for a bit of dirt here and there, you cannot really tell anything happened (the pages I was reading are somehow neatly stacked next to the hole). We looked in vain for somebody, so they can cover the hole.

“Good thing you are not superstitious,” Najuma tells me.


Woodin’s proof of the second incompleteness theorem for set theory

November 4, 2010

[Edit, Oct. 1, 2013: Robert Solovay has pointed out an inaccuracy in my presentation of Woodin’s argument: Rather than simply requiring that P is a hereditary property of models, we must require that \mathsf{ZFC} proves this. A corrected presentation of the argument will be posted shortly.]

As part of the University of Florida Special Year in Logic, I attended a conference at Gainesville on March 5–9, 2007, on Singular Cardinal Combinatorics and Inner Model Theory. Over lunch, Hugh Woodin mentioned a nice argument that quickly gives a proof of the second incompleteness theorem for set theory, and somewhat more. I present this argument here.

The proof is similar to that in Thomas Jech, On Gödel’s second incompleteness theorem, Proceedings of the American Mathematical Society 121 (1) (1994), 311-313. However, it is semantic in nature: Consistency is expressed in terms of the existence of models. In particular, we do not need to present a proof system to make sense of the result. Of course, thanks to the completeness theorem, if consistency is first introduced syntactically, we can still make use of the semantic approach.

Woodin’s proof follows.

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