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.

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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.

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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.

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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.)

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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 A\cos B-\cos A\sin 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 2\pi-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 2\pi.) 

Similarly: The direction of \vec u\times\vec v is obtained by the Right-Hand Rule, meaning \vec u\times \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 u\times \vec v is \|\vec u\|\|\vec v\|\sin\theta, where \theta is either the angle \alpha between \vec u and \vec v, or 2\pi-\alpha, whichever is between \null0 and \pi. 

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

Finally, \vec u\times\vec 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&0\end{array}\right|=(0,0,\cos A\sin B-\sin A\cos 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 2\pi, but the identity is true in all other cases as well, thanks to the periodicity properties of sine and cosine.)

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Set theory seminar -Forcing axioms and inner models II

September 21, 2008

In this second talk I proved the equivalence of Bagaria’s, and Goldstern-Shelah’s formulation of the bounded forcing axiom for a poset {\mathbb P} that preserves \omega_1.

We presented several characterizations of club subsets of {\mathcal P}_{\omega_1}(X) for X uncountable. We then defined when a forcing notion is proper and provided some basic examples of proper forcings, namely ccc, \sigma-closed forcings, and their products.

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Set theory seminar -Forcing axioms and inner models

September 12, 2008

Today I started a series of talks on “Forcing axioms and inner models” in the Set Theory Seminar. The goal is to discuss a few results about strong forcing axioms and to see how these axioms impose a certain kind of `rigidity’ to the universe.

I motivated forcing axioms as trying to capture the intuition that the universe is `wide’ or `saturated’ in some sense, the next natural step after the formalization via large cardinal axioms of the intuition that the universe is `tall.’

The extensions of {\sf ZFC} obtained via large cardinals and those obtained via forcing axioms share a few common features that seem to indicate their adoption is not arbitrary. They provide us with reflection principles (typically, at the level of the large cardinals themselves, or at small cardinals, respectively), with regularity properties (and determinacy) for many pointclasses of reals, and with generic absoluteness principles.

The specific format I’m concentrating on is of axioms of the form {\sf FA}({\mathcal K}) for a class {\mathcal K} of posets, stating that any {\mathbb P}\in{\mathcal K} admits filters meeting any given collection of \omega_1 many dense subsets of {\mathbb P}. The proper forcing axiom {\sf PFA} is of this kind, with {\mathcal K} being the class of proper posets. The strongest axiom to fall under this setting is Martin’s maximum {\sf MM}, that has as {\mathcal K} the class of all posets preserving stationary subsets of \omega_1.

Of particular interest is the `bounded’ version of these axioms, which, if posets in {\mathcal K} preserve \omega_1, was shown by Bagaria to correspond precisely to an absoluteness statement, namely that H_{\omega_2}\prec_{\Sigma_1}V^{\mathbb P} for any {\mathbb P}\in{\mathcal K}.

In the next meeting I will review the notion of properness, and discuss some consequences of {\sf BPFA}.

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275- Distance from a point to a plane

September 2, 2008

In class we tried to find the distance from P=(1,1,1) to the plane \Pi of equation x+y+5z=3.

There are several ways of doing this. For example:

  • Fix a point on the plane \Pi. Any point will do, say Q=(3,0,0).
  • Find a vector perpendicular to the \Pi. For example, \vec n=(1,1,5).

[Again, if a plane has equation Ax+By+Cz=D then \vec n=(A,B,C) is perpendiculart to it.

For example, x+y+5z=3 is parallel to the plane x+y+5z=0, which we can rewrite as (1,1,5)\cdot(x,y,z)=0, which is the equation of the set of points (x,y,z) perpendicular to the vector (1,1,5). This is to say, the plane x+y+5z=0 is perpendicular to the vector (1,1,5). Since the plane \Pi: x+y+5z=3 is parallel to x+y+5z=0, \Pi is also perpendicular to (1,1,5).

Another way of reaching the same conclusion is to rewrite Ax+By+Cz=D in the form A(x-x_0)+B(y-y_0)+C(z-z_0)=0 for some appropriate vector (x_0,y_0,z_0). There are many choices of (x_0,y_0,z_0) and they all work; all we need is that Ax_0+By_0+Cz_0=D, i.e., that (x_0,y_0,z_0) is in the original plane. For example, the point (3,0,0) belongs to the plane \Pi: x+y+5z=3, so we can rewrite the equation of \Pi as (x-3)+y+5z=0, which is equivalent to saying that (1,1,5)\cdot[(x,y,z)-(3,0,0)]=0. \null But this means that the vector \vec n=(1,1,5) is perpendicular to the vector (x,y,z)-(3,0,0), which is an arbitrary vector in the direction of the plane.]

Let’s continue with the problem of finding the distance from P to \Pi:

  • Consider the projection \vec w={\rm proj}_{\vec n}\overrightarrow{QP} of the vector \overrightarrow{QP} in the direction of \vec n. Clearly, the distance from P to \Pi is the length of \vec w.
  • Recall that {\rm proj}_{\vec u}\vec v=\frac{\vec u\cdot\vec v}{\|\vec u\|^2}\vec u.
  • Then \vec w=\frac{\overrightarrow {QP} \cdot \vec n}{\|\vec n\|^2}\vec n and the distance is \|\vec w\|=\frac{|\overrightarrow {QP} \cdot \vec n|}{\|\vec n\|}.
  • In detail, \overrightarrow{QP}=(-2,1,1) so \overrightarrow{QP}\cdot\vec n=-2+1+5=4 and \|\vec n\|=\sqrt{1+1+25}=\sqrt{27}, so \|\vec w\|=\frac4{3\sqrt3}.

Notice that (as discussed in class) the distance from a point P to the line in the direction of \vec v that goes through a point Q is given by d=\frac{\|\overrightarrow{QP}\times \vec v\|}{\|\vec v\|}, while (by the above) the distance from a point P to a plane containing a point Q and perpendicular to a vector \vec v is given by d=\frac{|\overrightarrow {QP} \cdot \vec v|}{\|\vec v\|}. While the expressions are similar, one involves a cross product and the other a dot product. This is because in one case we express the distance in terms of the sine of an angle \theta, and in the other, in terms of its cosine or, what is the same, in terms of the sine of \frac\pi2 -\theta. (Drawing a diagram may help you clarify the situation.)

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