### 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

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