# There are no P-points in Silver extensions

## TL;DR

The paper proves that Silver forcing extensions do not admit P-points, demonstrating the non-existence of P-points in models obtained via Silver and related forcings, and addresses open questions in set theory.

## Contribution

It establishes that Silver forcing prevents the extension of ground model ultrafilters to P-points and constructs models without P-points with large continuum.

## Key findings

- No P-points in Silver extensions.
- Silver forcing preserves the non-existence of P-points in further Sacks property extensions.
- Constructs models without P-points with arbitrarily large continuum.

## Abstract

We prove that after adding a Silver real no ultrafilter from the ground model can be extended to a P-point, and this remains to be the case in any further extension which has the Sacks property. We conclude that there are no P-points in the Silver model. In particular, it is possible to construct a model without P-points by iterating Borel partial orders. This answers a question of Michael Hru\v{s}\'ak. We also show that the same argument can be used for the side-by-side product of Silver forcing. This provides a model without P-points with the continuum arbitrary large, answering a question of Wolfgang Wohofsky.