Author: mpawliuk

Namba Forcing with Arnie Miller

(These are the voyages are the starship enterprise notes from Arnie Miller’s guest lecture in Alan Dow’s Forcing course, held on October 11, 2012. The lecture was a self-contained exploration of strange new worlds Namba forcing. (Alright, I will stop doing that. For those who don’t get the reference.) Arnie Miller’s quotes are in Grey. Yes, the forcing checks are ugly. I have yet to figure that one out.)

Summary

  • Motivation with posets that collapse cardinals.
  • Namba Trees are defined.
  • Two 0-extension lemmas are proved.
  • The Fat Namba Filter is introduced.
  • An important theorem of Namba’s is proved.

Continue reading Namba Forcing with Arnie Miller

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

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

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

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.

Continue reading Stevo’s Forcing Class Fall 2012 – Class 3

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} .

Continue reading Stevo’s Forcing Class Fall 2012 – Class 2