Tag: Combinatorial CH

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

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 .

Continue reading Stevo’s Forcing Class Fall 2012 – Class 6