### 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
- Genearlizations 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

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

# 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