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### 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

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

# Tag Archives: Large Cardinals

## A large cardinal in the constructible universe

In this post, we shall provide a proof of Silver’s theorem that the Erdos caridnal $\kappa(\omega)$ relativizes to Godel’s constructible universe. First, recall some definitions. Given a function $f:[\kappa]^{<\omega}\rightarrow \mu$, we say that $I\subseteq\kappa$ is a set of indiscernibles for … Continue reading

## On the consistency strength of the Milner-Sauer conjecture

Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading