### 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

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

# 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