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### 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

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

# 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