# Shelah’s Model without P-points– part 9

## Part 9: the main lemma ctd.

In short:

• Destroying P-points in any further $\omega^\omega$-bounding extensions.
• Second and final part of the proof.

Shelah's model without P-points part 9

Part 9 as PDF

Part 9 as Xournal-source

# Shelah’s Model without P-points– part 8

## Part 8: the main lemma

In short:

• Destroying P-points in any further $\omega^\omega$-bounding extensions.
• First part of the proof.

Shelah's model without P-points page 8

Part 8 as PDF

Part 8 as Xournal-source

# Shelah’s Model without P-points– part 7

My apologies for the blogging hiatus. Let’s continue.

## Part 7: switching filters

In short:

• From $u$ to $\{ \omega \} \otimes u$.

Shelah's model without P-points page 7

Part 7 as PDF

Part 7 as Xournal-source

# Shelah’s Model without P-points– part 6

## Part 6: properness ctd.

In short:

• Basic Properties:
• Grigorieff forcing and the Sacks variant are proper (part 2)

Shelah's model without P-points page 6

Part 6 as PDF

Part 6 as Xournal-source

# Shelah’s Model without P-points– part 5

## Part 5: properness

In short:

• Basic Properties:
• Grigorieff forcing and the Sacks variant are proper (part 1)

Shelah's model without P-points

Part 5 as PDF

Part 6 as Xournal-source

# Shelah’s Model without P-points– part 4

## Part 4: $\omega^\omega$-bounding

In short:

• Basic Properties:
• Grigorieff forcing and the Sacks variant are $\omega^\omega$-bounding

Shelah's model without P-points page 4

Part 4 as PDF

Part 4 as Xournal-source

# Shelah’s Model without P-points– part 3

## Part 3: More Strategy and Shelah’s “crucial fact”

In short:

• Basic Properties:
• a strategy for $\omega^\omega$-bounding — continued.
• The “crucial fact” for Grigorieff and Sacks forcing (that’s what Shelah calls it)

Shelah's model without P-points page 3

Part 3 as PDF

Part 3 as Xournal-source

# Shelah’s Model without P-points– part 1

To get some research back into this blog, I will, over the course of the next few weeks, post some handwritten notes about a couple of well-known results by Shelah: the model without P-points, the model with a unique selective ultrafilter and the model with a unique P-point.

These notes were written for my recent series of talks at the UofM’s Logic Seminar here in Ann Arbor and came out of an ongoing collaboration with David Chodounsky from Prague. The source for this presentation is Shelah’s Proper and Improper Forcing.

The notes were written using Xournal.

## Part 1: Basic Definitions

In short:

• Definitions
• Grigorieff forcing
• Sacks forcing with a filter
• P-filter, P-point
• non-meagre filter

Page 1

Part 1as PDF

Part 1 as Xournal-source