Stevo’s Forcing Class Fall 2012 – Class 9

(This is the ninth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the eighth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • We define the notion of properness.
  • Show that the poset \in -Collapse is proper.
  • Justin Moore gives some intuition.
  • Explain the Open Colouring Axiom.
  • Give a related proper poset.

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Stevo’s Forcing Class Fall 2012 – Class 8

(This is the eighth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the seventh lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. I didn’t take these notes, so there are few Stevo quotes. Thanks to Dana Bartasova for letting me reproduce her notes here. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • Show Chang’s Conjecture implies \theta_2 = \aleph_2 .
  • Show the failure of CC is equivalent to a nice colouring existing. This is fairly technical.
  • Introduce definable posets and “Definable CH”.

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Stevo’s Forcing Class Fall 2012 – Class 7

(This is the seventh lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the sixth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • Talk about the CH “hierarchy”.
  • Point out the MA analogue.
  • Show that it is possible to force a function to be continuous on a large set.
  • Show that Chang’s Conjecture is preserved by ccc forcing.

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Stevo’s Forcing Class Fall 2012 – Class 6

(This is the sixth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the fifth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • We talk about some caliber properties.
  • Establish that the ccc functor really goes to ccc spaces.
  • Use this to create an interesting ccc poset.
  • Talk about some combinatorial versions of CH.
  • Mention a few consequences of CH^3 .

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Stevo’s Forcing Class Fall 2012 – Class 5

(This is the fifth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the fourth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Sumary

  • Mention something about Cohen and Random Forcing.
  • We recall some facts about \theta_2 .
  • Establish some lemmas about \theta_2 .
  • Give most of the proof of the functor on ccc posets.

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Stevo’s Forcing Class Fall 2012 – Class 4

(This is the fourth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the third lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • We look at the real reason that regular open algebras don’t collapse \omega_1 to \omega .
  • Show there is a \sigma -linked, non \sigma -centred poset of size \mathfrak{p} .
  • Introduce the cardinal \theta_2 , related to Chang’s Conjecture.
  • Introduce the idea of a functor on ccc posets.

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Stevo’s Forcing Class Fall 2012 – Class 3

 (This is the third lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the second lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement.)

Summary

  • We look at 4 examples of posets that collapse \omega_1 to \omega .
  • A problem of Foreman is mentioned that asks about collapsing versus adding.
  • Talking about anti-Suslin trees.

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Stevo’s Forcing Class Fall 2012 – Class 2

(This is the second lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the first lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement.)

(Also, there are bound to be typos. I would appreciate anyone who could point them out in the comments and I will promptly fix them.)

Summary

  • Give the classic example of a \sigma -centred poset. (The one that adds a “small” subset of \omega .)
  • Define the various “Martin numbers”: \mathfrak{m}, \mathfrak{m}_{SH} and \mathfrak{m}_{sep} .
  • Define some small cardinals \mathfrak{p}, \mathfrak{t} and \mathfrak{b} .
  • Show that \mathfrak{m} = \mathfrak{m}_{SH} and \mathfrak{m}_{sep} = \mathfrak{p} .

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Is Algebra Necessary? – Andrew Hacker

Today on The Sunday Edition, on CBC Radio 1, host Michael Enright interviewed Andrew Hacker, about his controversial views on mathematics education. Basically, he thinks that demanding that all high school students master Algebra is “an onerous stumbling block for all kinds of students”. Listening to the interview this morning made me angry. I don’t get riled up easily, but my blood was boiling this morning.

Here is the link to the radio interview. (It starts around the 6:30 mark and is about 24 minutes long.) Here is the article Andrew Hacker wrote for the New York Times.

I intend to write a detailed response to the claims of Andrew Hacker later this week, but first let me share some reckless assertions that pissed me off. (These are paraphrases.)

“Should we be learning mathematics for learning’s sake? Don’t get me wrong, I’m a liberal-arts professor; I teach multi-disciplinary courses. However, I think that we should be focusing on learning about mathematics: the history of mathematics, the philosophy of mathematics. Not the dry stuff.”

“Hacker: 1.7 million Americans enter American universities every year, all required to have some high school mathematics. How many do you think choose to major in mathematics?

Micheal Enright: I don’t even want to guess.

Hacker: 15 Thousand. This tells me that even though teachers have had the chance to show the students the ‘beauty of mathematics’ they just aren’t getting it. [Goes on to imply that it’s because this is too hard to do.] Why are we training all of these students in high school when so few are going into STEM [science, technology engineering and mechanics] careers?”

“Do people need to learn about Fermat’s Theorem [sic] or Rye-man’s Hypothesis [sic]?”

I agree that something is broken with the way we learn, teach and perceive mathematics, especially in middle years and high school. This is a conversation we must have. However, it is troubling when the social scientists try to use their irrelevant authority to persuade the masses, leveraging the common man’s bias against mathematics.

I’m mad. I don’t like when people piss on what I love, and I especially dislike when they do it while tricking the public. As people we need emotions: love, anger, joy, contempt; but as a people we need thought, analysis, reason and mathematics to universally prevail.

Again, I plan on writing a cool-headed, but strong reply, and this is just me venting. Until then, let me leave you with this, taken from JFK’s famous speech:

“But why, some say, the moon? Why choose this as our goal? And they may well ask why climb the highest mountain? Why, 35 years ago, fly the Atlantic? Why does Rice play Texas? We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.”

Stevo’s Forcing Class Fall 2012 – Class 1

(In the fall of 2012 I will be taking Stevo Todorcevic’s class in Forcing at the University of Toronto. I will try to publish my notes here, although that won’t always be possible.)

Summary of class 1.

  • Discuss examples of ccc posets whose product is not ccc.
  • Prove a theorem of Baumgartner’s that relates the branches in a tree to its antichain structure.
  • Display the differences in chain conditions.

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