### 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

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

# 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