### 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

Martin's Axiom PFA Hedetniemi's conjecture Almost-disjoint famiy Knaster Foundations Successor of Singular Cardinal xbox Non-saturation Uniformly coherent Commutative cancellative semigroups Chromatic number Parameterized proxy principle weak square Shelah's Strong Hypothesis weak diamond OCA Antichain reflection principles Rado's conjecture Singular coﬁnality Poset HOD Microscopic Approach Hindman's Theorem Postprocessing function Slim tree Jonsson cardinal diamond star Kurepa Hypothesis L-space Weakly compact cardinal Small forcing Successor of Regular Cardinal Forcing Reduced Power Prevalent singular cardinals Luzin set Square-Brackets Partition Relations P-Ideal Dichotomy free Boolean algebra Souslin Tree Cohen real middle diamond Generalized Clubs Universal Sequences Rainbow sets Cardinal Invariants Erdos-Hajnal graphs b-scale polarized partition relation Fast club 05A17 20M14 Ascent Path coloring number 05D10 Dushnik-Miller Constructible Universe Minimal Walks stationary reflection Prikry-type forcing Hereditarily Lindelöf space sap Fat stationary set Diamond Nonspecial tree Cardinal function Fodor-type reflection Mandelbrot set Coherent tree Whitehead Problem Aronszajn tree tensor product graph Stevo Todorcevic Axiom R Ostaszewski square Distributive tree Erdos Cardinal stationary hitting 11P99 square PFA(S)[S] very good scale projective Boolean algebra S-Space Uniformization Absoluteness Large Cardinals Chang's conjecture square principles Almost countably chromatic Sakurai's Bell inequality approachability ideal Almost Souslin Forcing Axioms Partition Relations Club Guessing Singular Density ccc incompactness Selective Ultrafilter Singular cardinals combinatorics Rock n' Roll

# 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