Tag: forcing

Future of Set Theory – Panel Discussion

The following is a (somewhat complete) transcription of a panel discussion held at the Fields Institute on April 1, 2015 as part of the “Forcing and its Applications Retrospective” workshop.

The speakers were

  • Stevo Todorcevic (University of Toronto)
  • Jindrich Zapletal (University of Florida)
  • Christina Brech (University of São Paulo)
  • Assaf Rinot (Bar-Ilan University)
  • Matteo Viale (University of Torino)
  • Justin Moore (Cornell University, Moderator)

I took notes, so please bear in mind that some of this is paraphrased. I apologize if I misrepresented anyone, and I am happy to make corrections if I am emailed.

Continue reading Future of Set Theory – Panel Discussion

Stevo’s Forcing Class Fall 2012 – Class 15

(This is the fifteenth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the fourteenth 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 introduce free sequences.
  • A lemma (2) relating free sequences and \pi -bases is proved.
  • We prove that the generic ultrafilter is countably generated.
  • We construct nice \sigma -disjoint \pi -bases.
  • A realted theorem about compact spaces supporting a radon measure is proved as a corollary.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 15

Stevo’s Forcing Class Fall 2012 – Class 14

(This is the fourteenth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the thirteenth 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 spend the entire lecture proving a theorem about Baire-Class-1 functions.
  • The proof is interesting because it uses forcing.
  • The entire proof is in the inductive step.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 14

Stevo’s Forcing Class Fall 2012 – Class 12

(This is the twelfth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the eleventh 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 show that PFA implies the Weak Baumgartner Axiom.
  • With \mathfrak{p} > \omega_1 the weak BA is equivalent to BA.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 12

Stevo’s Forcing Class Fall 2012 – Class 13

(This is the thirteenth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the twelfth 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 restate the strong lemma.
  • A handful of spaces are presented to illustrate the following theorem.
  • A theorem about Baire-Class-1 functions is stated. (Which is proved by forcing!)
  • The proof strategy is discussed.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 13

Stevo’s Forcing Class Fall 2012 – Class 11

(This is the eleventh lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the tenth 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

  • Baumgartner’s Axiom is introduced.
  • A combinatorial lemma is proved, which really shows that a certain poset is proper.
  • Some words are said about the productivity of chain conditions.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 11

Stevo’s Forcing Class Fall 2012 – Class 10

(This is the tenth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the ninth 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. I am sure that this one has a ton of typos. There are also some omissions which I will try to patch up.)

Summary

  • We define the Side Condition Method.
  • We force the P-Ideal Dichotomy.
  • A cardinal preserving modification of the side condition is discussed.
  • Semi-properness is introduced, and then used to prove a fact about projectively stationary sets.
  • The Strong Reflection Principle is introduced.
  • Some consequences are discussed.
  • A ZFC version of diamond is stated.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 10

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.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 9

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”.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 8

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.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 7