Tag: Proper Forcing

Stevo’s Forcing Class Fall 2012 – Class 16

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

  • This lecture introduces the idea of forcing things that force things to be true.
  • Five different types of these theorems are stated.
  • We prove one of them. Namely that assuming PFA, after forcing with Sacks, MA(\aleph_1) is true.

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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 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 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