### Archives

### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

### Keywords

Almost countably chromatic Knaster middle diamond Ostaszewski square Universal Sequences Nonspecial tree Partition Relations square Small forcing Erdos Cardinal Aronszajn tree Large Cardinals 20M14 tensor product graph Cardinal function ccc Sakurai's Bell inequality Reduced Power very good scale Postprocessing function Coherent tree Club Guessing Cohen real free Boolean algebra Erdos-Hajnal graphs Chang's conjecture Singular Density Martin's Axiom Mandelbrot set Luzin set Forcing Axioms 11P99 Square-Brackets Partition Relations Non-saturation Hindman's Theorem weak diamond weak square xbox Forcing b-scale Whitehead Problem Souslin Tree Kurepa Hypothesis super-Souslin tree Almost Souslin Fat stationary set Minimal Walks Rock n' Roll Successor of Regular Cardinal Rado's conjecture Hedetniemi's conjecture L-space Foundations Singular cardinals combinatorics Generalized Clubs PFA Jonsson cardinal Parameterized proxy principle Prevalent singular cardinals Almost-disjoint famiy Absoluteness Fast club Chromatic number Microscopic Approach Diamond diamond star Distributive tree Dushnik-Miller S-Space square principles approachability ideal 05A17 incompactness HOD Axiom R Cardinal Invariants Uniformly coherent Selective Ultrafilter stationary hitting Ascent Path Successor of Singular Cardinal Fodor-type reflection projective Boolean algebra polarized partition relation Poset Stevo Todorcevic Constructible Universe PFA(S)[S] Antichain reflection principles Singular coﬁnality OCA Weakly compact cardinal Slim tree Uniformization Hereditarily Lindelöf space Commutative cancellative semigroups coloring number Rainbow sets sap Prikry-type forcing P-Ideal Dichotomy Shelah's Strong Hypothesis stationary reflection

# Tag Archives: PFA

## Square principles

Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading

## PFA and the tree property at $\aleph_2$

Recall that a poset $\langle T,\le\rangle$ is said to be a $\lambda^+$-Aronszajn tree, if it isomorphic to a poset $(\mathcal T,\subseteq)$ of the form: $\emptyset\in \mathcal T\subseteq{}^{<\lambda^+}\lambda$; Write $\mathcal T_\alpha:=\{\sigma\in\mathcal T\mid \text{dom}(\sigma)=\alpha\}$; for all $\alpha<\lambda^+$, $\mathcal T_\alpha$ has size $\le\lambda$, … Continue reading