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P-measures in models without P-points

Piotr Borodulin-Nadzieja, Jonathan Cancino-Manríquez, Adam Morawski

Abstract

We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain unique P-point model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by $ω$ product of Silver forcings and that in the model obtained by the product of $ω_2$ many Silver forcings there are no P-measures of countable Maharam type

P-measures in models without P-points

Abstract

We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain unique P-point model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by product of Silver forcings and that in the model obtained by the product of many Silver forcings there are no P-measures of countable Maharam type
Paper Structure (9 sections, 43 theorems, 87 equations)

This paper contains 9 sections, 43 theorems, 87 equations.

Table of Contents

  1. P-points and P-measures
  2. Measures on $\omega$
  3. Silver forcing
  4. $\omega^\omega$-bounding.
  5. Interval partitions
  6. Measures which cannot be extended to P-measures
  7. Rapid Filters and P-measures
  8. More on P-measures under CH
  9. A model with P-measures and without P-points

Key Result

Theorem 3

It is consistent with $\mathsf{ZFC}$ that there is a P-measure but there is no P-point.

Theorems & Definitions (83)

  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 1.1
  • Proposition 1.2
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • ...and 73 more