Set theory seminar -Forcing axioms and inner models -Intermezzo

September 30, 2008
This posting complements a series of talks given at the Set Theory Seminar at BSU from September 12 to October 24, 2008. Here is a list of links to the talks in this series:
  • First talk, September 12, 2008.
  • Second talk, September 19, 2008.
  • Third talk, September 26, 2008.
  • Fourth talk, October 3, 2008.
  • Fifth talk, October 10, 2008.
  • Sixth talk, October 17, 2008.
  • Seventh talk, October 24, 2008.

[Version of October 31.]

I’ll use this post to provide some notes about consistency strength of the different natural hierarchies that forcing axioms and their bounded versions suggest. This entry will be updated with some frequency until I more or less feel I don’t have more to add. Feel free to email me additions, suggestions and corrections, or to post them in the comments. In fact, please do.

Read the rest of this entry »

Advertisements

580- Topics in Set Theory. ANNOUNCEMENT

September 30, 2008

This Spring I will be teaching Topics in set theory. The unofficial name of the course is Combinatorial Set Theory. 

We will cover diverse topics in combinatorial set theory, depending on time and the interests of the audience, including partition calculus (a generalization of Ramsey theory), cardinal arithmetic, and infinite trees. Time permitting, we can also cover large cardinals, determinacy and infinite games, or cardinal invariants (the study of sizes of sets of reals), among others. I’m open to suggestions for topics, so feel free to email me or to post in the comments. 

Prerequisites: Permission by instructor (that is, me).

Recommended background: Knowledge of cardinals and ordinals. A basic course on set theory (like 502: Logic and Set Theory) would be ideal but is not required.

The course may be cancelled if not enough students enroll, which would make us all rather unhappy, so don’t let this happen.


175, 275 -Homework 5

September 30, 2008

Homework 5 is due Tuesday, October 7, at the beginning of lecture. Same remarks as before apply.

175: Section 7.1, exercises 20, 30, 32, 42.
Section 7.2, 40.
Section 9.3, exercise 23.
Each problem is worth 2 points; there are 2 extra credit points.

275: Section 12.1, exercises 13-18, 44.
Section 12.2, exercises 36, 54.
Section 12.3, exercises 21, 22, 30.
The exercises in section 12.3 are worth 1 point each, the combined exercise 12.1.13-18 is worth 3 points, the other exercises are worth 2 points each; there are two extra credit points.


175, 275 – Midterm 1

September 30, 2008

The first exam was Friday, September 26, during lecture. This exam is worth 10% of the total grade. Books, notes, and calculators were allowed. 

Failure to take the exam was graded as a score of 0. 

175: The exam covered Chapters 6 and 9 of the textbook (and assumed knowledge of Calc I). Exam, Graph.

275: The exam covered Chapters 10 and 11 of the textbook (and assumed knowledge of Calc I and II). Exam, Graph. (Typo: In problem 2.d, the equation of the plane must be x+y-2z=2.)


Youth without youth

September 23, 2008

Youth without youth is director Francis Ford Coppola’s most recent film (2007), the first since The Rainmaker (1997).

What a disappointment! The story is impossibly meandering and most of it ends up not going anywhere. For example, for an hour or so, we are in the middle of a Nazi conspiracy, only to have it forgotten as we move twenty or so years into the future. The way this is handled, most of what happened with the Nazis turns out to be irrelevant. The film looks beautiful, but there is no narrative to guide it. There are some rather pedestrian conventions as well, which made me wince whenever they were used.

The main character, Dominic, is a philosopher of language interested in the origin of language, in some mythical protolanguage from which all others sprang. This idea of a single universal origin for language is already bothersome, but let’s say he really only means a proto-Indoeuropean language, which still fails to be convincing anyway. Unsatisfied with his life, lonely and sad for a love lost many years ago (not that the film tries to emotionally connect the audience), he decides to kill himself.

Before going through with his plan, light strikes and he ends up in a hospital, basically reborn. It is Easter, just in case we miss the subtlety. Instead of dying, the lightning somehow induces a rejuvenation process. It also gives him superpowers.

Sigh.

Not only superpowers, actually. It also gives him a “double,” that we see in mirrors, and who may or may not be evil, not that it matters or we care. Nazis blah blah OSS blah. At the end, he kills the double by breaking a mirror. There is shrieking.

His double is not his first victim. Before, he kills a Nazi bad guy with his magic mental powers. Before, any hope one might have had for this film had died as well.

Oh, there are dreams throughout the film. For no good reason, dreams are shown upside down. Whenever we see an upside down sequence (much more often than one would like), we are witnessing a dream. It is a plot device. At the end of the film, there is a scene which we would have clearly identified as a dream except that it is not upside down. So we know it is not a dream but New Age mystical magic. There is a lot of silly New Age mystical magic throughout the film.

After the Nazi nonsense, the movie turns into an exploration of Indian mysticism. Dominic finds his long lost love reincarnated. The woman starts to regress, and he moves in with her, and uses her regressions to explore his protolanguage theory. Oh, but she will die if he continues, so he doesn’t. Another plot that ends up going nowhere.

Tim Roth plays Dominic. Roth is in his mid-forties. People throughout the film, even people noticeably younger, call him a young man. I don’t know, maybe they see his soul, this movie is that silly.


Arbiter

September 23, 2008

The Arbiter, the student paper at BSU, had this Monday a short note on new faculty, including excerpts of a brief interview with myself.


275- A problem from Homework 2

September 21, 2008

Quite a few of you had difficulties with problem 18 of the Additional and Advanced Exercises for Chapter 10, so I am posting a solution here.  

The problem asks to derive the trigonometric identity sin(A-B)=sin Acos B-cos Asin B by forming the cross product of two appropriate vectors.

In problems that involve trigonometry or geometry, it is convenient to begin with vectors that have some clear geometric meaning related to the problem at hand, so it seems natural to consider the vectors vec u=(cos A,sin A,0) and vec v=(cos B,sin B,0).

These are two vectors in the plane, but we look at them as vectors in 3-D, as we should, since we want to look at their cross product, and this is only defined for vectors in 3-D.

So: vec u is a vector of size 1 (|vec u|=sqrt{cos^2 A+sin^2 A+0^2}=1) that forms an angle of A radians with the x-axis (measured counterclockwise). Similarly, vec v is a vector of size 1 that forms an angle of B radians with the x-axis (measured counterclockwise).

Now: We need to analyze the angle between vec u and vec v, which seems to be the technical point of this exercise, so let’s do this very carefully. This angle is the angle measured starting at vec u and moving counterclockwise until we find vec v, is usually B-A, but it may be 2pi-A+B if, for example, vec u and vec v are vectors in the first quadrant and B<A. (Although we can “ignore” this case since the sine function is periodic with period 2pi.) 

Similarly: The direction of vec utimesvec v is obtained by the Right-Hand Rule, meaning vec utimes vec v is a vector perpendicular to the plane spanned by vec u and vec v (the xy-plane), but it may be a positive (or zero) multiple of {bf k}, or a negative multiple of {bf k}, depending on whether the angle between vec u and vec v is smaller than (or equal to) pi, or larger than pi.

The magnitude of vec utimes vec v is |vec u||vec v|sintheta, where theta is either the angle alpha between vec u and vec v, or 2pi-alpha, whichever is between null0 and pi. 

Putting these two bits of information (about direction and magnitude) together, we find that vec utimesvec v=(0,0,sin(B-A)) if 0le B-Alepi. If B-A>pi, then vec utimesvec v=(0,0,-sin(2pi+A-B) but sin(2pi+alpha)=sin(alpha)=-sin(-alpha) for any alpha, so also in this case vec utimesvec v=(0,0,sin(B-A)).

Finally, vec utimesvec v, component-wise, is found by computing the formal determinant left|begin{array}{ccc}{bf i}&{bf j}&{bf k}\ cos A&sin A&0\ cos B&sin B&0end{array}right|=(0,0,cos Asin B-sin Acos B). Comparing this expression with the one above, we find the desired identity.

(Actually, we find it with the roles of A and B reversed, but this is of course irrelevant. And of course this deduction only works for angles between null0 and 2pi, but the identity is true in all other cases as well, thanks to the periodicity properties of sine and cosine.)