Disqus
I have just enabled Disqus commenting system. It has lots of nice features, but there are a few bugs we need to fix — that’s part of the Boole’s Rings way… The most obvious issue is that MathJax...
View ArticleA partition theorem for finite trees
[Could not find the bibliography file(s) In a recent paper [?], Steven Gubkin, Daniel McDonald, Manuel Rivera, and I stumbled across what appears to be a new result in Ramsey theory. As the title of...
View ArticleConvergence of ideas
As a moderator on MathOverflow, I see a lot of interesting interactions between mathematicians. The occasional dramatic situations get discussed profusely by community but very few take the time talk...
View ArticlePossibly true. Necessarily funny.
This is the tagline for fauxphilnews, a hilarious philosophy blog created by Ben Bronner. My favorite entry so far is Saul Kripke’s resignation after faking results of thought experiments. Check it out…
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Bumper sticker
An amazing idea for a bumper sticker from xkcd: Of course, this has many implications for people driving behind you…
View ArticleEnvelope forcing
In a recent paper [1], Jared Corduan and I considered various notions of combinatorial indecomposability for finite ordinal powers of \(\omega.\) In this process, we uncovered two weak forms of...
View ArticleBack to the origin…
I am now settling into my new place in Hanover, New Hampshire. I am starting my new position as a John Wesley Young Research Instructor at Dartmouth College on July 1. It’s great to be back where it...
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SMBC on madness…
It’s very easy to imagine a mad scientist: combine a bad hair day with a lab coat, surround with vats, oscillators, and other instruments, throw the mix into a cave and voilà! It’s much harder to...
View ArticleDiaconescu’s Theorem
In the 1970s, Radu Diaconescu [1] showed that the Axiom of Choice implies the Principle of Excluded Middle. More specifically, he showed that every topos that satisfies (a very mild form of) the Axiom...
View ArticleScheming schemes…
In everyday language, the word scheme often has a negative connotation: scheme is used as a synonym for devious plan. In mathematical language, the word has no negative aspect at all. In fact, I think...
View ArticleBack to Cantor?
Set Theory has a fantastic and legendary history. At the end, it left us with ZFC, which is currently recognized as the foundation of mathematics. This state of affairs is arguably one of the best...
View ArticleTowsner’s stable forcing
It is well known that a model $\newcommand{\MN}{\mathfrak{N}}\MN$ of $\newcommand{\RCA}[1]{\mathsf{RCA}_{#1}}\RCA0$ satisfies $\Sigma^0_n$-induction...
View ArticleArithmetical consequences of the set-theoretic multiverse
In [2], Joel David Hamkins proposed a set of axioms for the set-theoretic multiverse. Several of these axioms reflect the typical world many set theorists live in, namely that generic extensions, inner...
View ArticleWhat is combinatorial set theory?
This is a very difficult question that I find myself pondering every so often. I once heard a story that Jim Baumgartner was asked this question at a major logic meeting where all areas — recursion...
View ArticleSelected Papers Network
I just made my first contribution to the Selected Papers Network. It was fun and easy and I strongly recommend you use it too! It’s too early for serious commentary on the experience but there are a...
View ArticleHomotopy Type Theory
After a productive year at the Institute for Advanced Study, the Univalent Foundations Program has written a book on Homotopy Type Theory (HoTT). The foreword gives a succinct description of the...
View ArticleHoTT Math Series
I am planning to do a series of posts where I attempt to do math in Homotopy Type Theory (HoTT). The plan is to do some relatively simple proof-relevant mathematics at an informal level. The topics...
View ArticleHoTT Math 1: Elementary group theory
This is the first in a series of posts on doing mathematics in Homotopy Type Theory (HoTT). Various conventions and notations are explained in the preamble; there are no additional prerequisites for...
View ArticleHoTT Math 2: More on equational logic
Last time, I promised we would look at fields. I have to delay this by one or two installments of HoTT Math since there is so much to say and I am struggling to keep these short. This edition of HoTT...
View ArticleHoTT Math 3: Unit group of a ring
In this installment of HoTT Math, we are taking one more step toward elementary field theory by exploring the unit group of a ring. A commutative (unital) ring is a set \(R\) with two constants...
View ArticleHoTT Math 4: Local rings and fields
It’s been a while since the last edition of HoTT Math. Fall is always very busy for me and I’ve been composing this installment one \(\varepsilon\) at a time… We are finally arriving at our...
View ArticleOn the structure of universes in HoTT
In this post, I want to outline some subtle and interesting aspects and issues with respect to universes in HoTT. Some of these issues were brought up in a comment discussion with Mike Shulman some...
View ArticleSuper HoTT
This is a follow-up to my recent post on the structure of universes in HoTT. In this post I will outline one of the possible alternative ways of handling universes in HoTT, which I will colorfully call...
View ArticleSome Cardinal Arithmetic
Some time ago, Asaf Karagila wrote wonderful post wherein he shows that, even without assuming the axiom of choice one can always find four cardinals \(\mathfrak{p} \lt \mathfrak{q}\) and...
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