### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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