Set theory seminar -Forcing axioms and inner models VII

October 24, 2008

I concluded my series of talks by showing the following theorem of Viale:

Theorem (Viale). Assume {\sf CP}(\kappa^+) and let M\subseteq V be an inner model where \kappa is regular and such that (\kappa^+)^M=\kappa^+. Then {\rm cf}(\kappa)\ne\omega.

This allows us to conclude, via the results shown last time, that if {\sf PFA} holds in V and M computes cardinals correctly, then it also computes correctly ordinals of cofinality \omega.

An elaboration of this argument is expected to show that, at least  if we strengthen the assumption of {\sf PFA} to {\sf MM}, then M computes correctly ordinals of cofinality \omega_1.

Under an additional assumption, Viale has shown this:  If {\sf MM} holds in V, \kappa is a strong limit cardinal, (\kappa^+)^M=\kappa^+, and in M we have that \kappa is regular, then in V the cofinality of \kappa cannot be \omega_1. The new assumption on \kappa allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and I_{<f} [\lambda], Fund. Math. 148 (1995), 165-198, in place of the structure imposed by {\sf CP}(\kappa^+). It is still open if the corresponding covering statement {\sf CP}(\kappa^+,\omega_1) follows from {\sf MM}, which would eliminate the need for this the strong limit requirement.

  • Go to the intermezzo for a discussion of consistency strengths.

275- Constant curvature

October 23, 2008

The text we are using for Calculus III introduces the notion of unit tangent vector, principal unit normal vector, and curvature, for smooth curves \vec r(t), and it also mentions that circles and lines are planar curves of constant curvature. (A curve is planar if its image is contained in a plane, i.e., if it describes a two-dimensional trajectory.)

Surprisingly, though, the text does not explain (not even in the exercises) that circles and lines are the only smooth planar curves of constant curvature. The argument for this is simple enough, so I will show it here:

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Partitioning numbers

October 22, 2008

A student asked me the other day the following rather homework-looking question: Given a natural number n, how many solutions (x,y) does the equation


have for x and y natural numbers?

The question has a very easy answer: Simply notice that 0\le 2y\le n and that any y like this determines a unique x such that (x,y) is a solution. So, there are \frac n2 +1 solutions if n is even (as y can be any of 0,1,\dots,n/2), and there are \frac{n+1}2 solutions if n is odd.

I didn’t tell the student what the answer is, but I asked what he had tried so far. Among what he showed me there was a piece of paper in which somebody else had scribbled


which caught my interest, and is the reason for this posting.

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175, 275 -Suggestions for next week.

October 21, 2008

Remember that on October 31 is the second midterm. There will not be a homework set due on November 4. However, we will cover new material during the final week of October. If you want to read ahead, in 175 we will be covering the beginning of Chapter 8, probably up to section 8.3. In 275, we will be covering the beginning of Chapter 13; I doubt we will get to section 13.4, but you may want to start reading the first 4 sections of this chapter. 

The exam will concentrate on material covered after the first midterm, but it is cumulative. For 175, it will include the basics of Taylor’s theorem (and it may be a good idea to take a look at Talman’s paper; see here); but it won’t include the proof that the method of partial fractions decomposition works. For 275, you may want to review the polar expression of the Laplacian and how to derive it. For both courses, a safe assumption is that if something was covered in lecture (up to October 24), then it may be included.

175, 275 -Homework 8

October 21, 2008

Homework 8 is due Tuesday, October 28, at the beginning of lecture. Same remarks as before apply.

175: Section 7.7, exercises 11, 20, 32, 42, 50, 65, 74.
Section 8.1, exercises 2, 8, 11, 12, 16, 42, 49, 81.
Solve as many problems as you want; of those, up to 12 will be chosen randomly and graded. Each exercise is worth 1 point, so you may obtain two extra credit points.

275: Section 12.7, exercises 2, 20, 36, 40, 44, 46.
Section 12.9, exercises 4, 10, 12.
Section 13.1, exercises 6, 24.
This homework will be graded out of 10 points. Each exercise is worth 1 point. You may obtain one extra credit point.

175 -Pi and Simpson's method

October 20, 2008

Here is a .pdf file of a Maple worksheet showing how Simpson’s method can be used to compute \pi, following the exercise I suggested in lecture. The accuracy I obtained is larger than what I asked for in lecture, at the cost of a large number of intervals being required.

There are of course much faster methods to compute \pi, based on other ideas.

Set theory seminar -Forcing axioms and inner models VI

October 17, 2008

I presented a sketch of a nice proof due to Todorcevic that {\sf PFA} implies the P-ideal dichotomy {\sf PID}. I then introduced Viale’s covering property {\sf CP} and showed that it follows from {\sf PID}. Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:

Theorem (Viale). Assume M\subseteq V is an inner model.

  1. If {\sf PFA} holds in V and M computes cardinals correctly, then it also computes correctly ordinals of cofinality \omega.
  2. If {\sf MM} holds in V, \kappa is a strong limit cardinal, (\kappa^+)^M=\kappa^+, and in M we have that \kappa is regular, then in V, the cofinality of \kappa cannot be \omega_1.

It follows from this result and the last theorem from last time that if V is a model of {\sf MM} and a forcing extension of an inner model M by a cardinal preserving forcing, then {\sf ORD}^{\omega_1}\subset M.

In fact, the argument from last time shows that we can weaken the assumption that V is a forcing extension to the assumption that for all \kappa there is a regular cardinal \lambda\ge\kappa such that  in M we have a partition S^\lambda_\omega=\sqcup_{\alpha<\kappa}S_\alpha where each S_\alpha is stationary in V.

It is possible that this assumption actually follows from {\sf MM} in V. However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable V cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of V that preserves cardinals, does not add reals, and (for some cardinal \kappa) the set of points of countable V-cofinality in \lambda is nonstationary for every regular \lambda\ge\kappa^+. Obviously, this situation is incompatible with {\sf PFA} in V, by Viale’s result.

Graduate student seminar – Very large finite numbers

October 15, 2008

Today I gave a talk at the Graduate student seminar on some recent combinatorial results that produce sequences that grow very fast (in a precise sense). The talk is titled Very large finite numbers and here is the .pdf file.

It is based on two recent papers of mine that can be found on my papers page: Regressive functions on pairs, that has been submitted to the European Journal of Combinatorics, and Goodstein’s function, that appeared in Revista Colombiana de Matemáticas 41 (2)(2007), 381-391.

For a nice introduction to incompleteness and Goodstein’s theorem, see Will Sladek’s paper here.

175 -Error from Simpson's rule

October 15, 2008

I am following the nice article “Simpson’s rule is exact for quintics” by Louis A. Talman, in its computation of the error term for approximating integrals using Simpson’s rule. The article was published in the American Mathematical Monthly, vol 113 February 2006, pp. 144-155. Albertsons library is subscribed to the Monthly, so you can find the article there. You can also find it in Dr. Talman’s page. More details on Taylor’s formula can be found on section 8.9 of the book.

175, 275 -Homework 7

October 14, 2008

Homework 7 is due Tuesday, October 21, at the beginning of lecture. Same remarks as before apply.

175: Section 7.4, exercise 50.
Section 7.6, exercises 6, 10, 24, 27, 34.
Practice Exercises for Chapter 7, exercises 46, 48, 98.
Section 8.9, exercises 41, 42, 44.
Each exercise is worth 1 point, and you may obtain two extra credit points.

275: Section 12.5, exercises 6, 11, 16, 24, 28.
Section 12.6, exercises 2, 6, 14, 18, 40, 56, 57.
This homework will be graded out of 10 points. Each exercise is worth 1 point. You may obtain two extra credit points.