Tag: MA

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

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.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 5

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.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 4

Another Combinatorial Result

Here is Chris Eagle’s presentation from Stevo Todorcevic’s class “Combinatorial Set Theory”.

From the abstract:

We prove that MA + \mathfrak{c} = \aleph_2 implies \mathfrak{c} \not\rightarrow (\mathfrak{c}, \omega+2)^2 . The exposition is based on hand-written notes provided by S. Todorcevic. The result itself is due to R. Laver.

This is the analogous result to “MA implies (NonSpecial Tree) \not\rightarrow (NonSpecial Tree, \omega+2)^2 “, which I explained here.