A forcing axiom for a non-special Aronszajn tree
Abstract
Suppose that $T^*$ is an $\omega_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than $\omega_1$, and the $P$-ideal dichotomy. On the other hand, PFA($T^*$) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2018
- DOI:
- 10.48550/arXiv.1805.08164
- arXiv:
- arXiv:1805.08164
- Bibcode:
- 2018arXiv180508164K
- Keywords:
-
- Mathematics - Logic;
- 03E35;
- 03E57