### 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

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

# 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