Asaf Karagila
I don't have much choice...

On the Partition Principle

Last Wednesday I gave a talk about the Partition Principle in our students seminar. This talk covered the historical background of the oldest open problem in set theory, and two proofs that for a long time I avoided learning. I promised to post a summary of the talk here. So here it is. The historical data was taken from the paper by Banaschewski and Moore, "The dual Cantor-Bernstein theorem and the partition principle." (MR1072073) as well Moore's wonderful book "Zermelo’s Axiom of Choice" (which has a Dover reprint!).

To Colloops a cardinal

This is nothing new, but it's a choice-y way of thinking about it. Which is really what I enjoy doing.

Definition. Let $V$ be a model of $\ZFC$, and $\PP\in V$ be a notion of forcing. We say that a cardinal $\kappa$ is "colloopsed" by $\PP$ (to $\mu$) if every $V$-generic filter $G$ adds a bijection from $\mu$ onto $\kappa$, but there is an intermediate $N\subseteq V[G]$ satisfying $\ZF$ in which there is no such bijection, but there is one for each $\lambda\lt\kappa$. Continue reading...

Huge cardinals are huge!

In a previous post, I gave a humorous classification of large cardinals, dividing them to large large cardinals and small large cardinals, and so on. In particular huge cardinals were classified as large large large large large cardinals. But how large are they? Not surprisingly, very large.

In case you forgot, $\kappa$ is a huge cardinal if there is an elementary embedding $j\colon V\to M$, where $M$ is a transitive class containing all the ordinals, with $\kappa$ critical, and $M$ is closed under sequences of length $j(\kappa)$. Continue reading...

Anti-anti Banach-Tarski arguments

Many people, more often than not these are people from analysis or worse (read: physicists, which in general are not bad, but I am bothered when they think they have a say in how theoretical mathematics should be done), pseudo-mathematical, non-mathematical, philosophical communities, and from time to time actual mathematicians, would say ridiculous things like "We need to omit the axiom of choice, and keep only Dependent Choice, since the axiom of choice is a source for constant bookkeeping in the form of non-measurable sets".

People often like to cite the paradoxical decomposition of the unit sphere given by Banach-Tarski. "Yes, it doesn't make any sense, therefore the axiom of choice needs to be omitted". Continue reading...

Ramsey cardinals are large large small large cardinals

There is no well defined notion for what is a large cardinal. In some contexts those are inaccessibles, in others those are critical points of elementary embeddings, and sometimes $\aleph_\omega$ is a large cardinal.

But we can clearly see some various degrees of largeness by how much structure the existence of the cardinal imposes. Inaccessible cardinals prove there is a model for second-order $\ZFC$, and Ramsey cardinals imply $V\neq L$. Strongly compact cardinals even imply that $\forall A(V\neq L[A])$. Continue reading...

My love-hate relationship with forcing

Forcing is great. Forcing is an amazing method. If you can think about it, then you can probably force to make it happen. All it requires is some creativity and rudimentary understanding of the objects that you are working with.

Forcing is horrible. If you can think about it, you can encode it into generic objects. If you can't think about it, you can encode it into generic objects. If you think that you can't encode it into generic objects, then you are probably wrong, and you can still encode it into generic objects. Continue reading...

How Fields Became "Nobel"

Here is some interesting piece of mathematical history: How the Fields medal went from "Soviet award" to "Mathematical Nobel".

This is not a blog post.

This is not a blog post.

No uniform ultrafilters

Earlier this morning I received an email question from Yair Hayut. Is it consistent without the axiom of choice, of course, that there are free ultrafilters on the natural numbers but none on the real numbers?

Well, of course that the answer is negative. If $\cal U$ is a free ultrafilter on $\omega$ then $\{X\subseteq\mathcal P(\omega)\mid X\cap\omega\in\cal U\}$ is a free ultrafilter on $\mathcal P(\omega)$. But that doesn't mean that the question should be trivialized. What Yair asked was actually slightly subtler than that: is it consistent that there are free ultrafilters on $\omega$, but no uniform ultrafilters on the real numbers? Continue reading...

Why Carl Sagan was better than Neil deGrasse Tyson, and from the most of us too

I've recently watched the finale of Cosmos, the new version, presented by Neil deGrasse Tyson. It was a very nice series which seem to push forward the fact that science is based on not knowing, rather than knowing, and the will to know. No, not will, the need to know. We need to know, and this is why we go on searching the answers to questions that haunt us.

Neil deGrasse Tyson pushed a lot on the point that we really push the planet to its limits, and we might be close to the point of no return from which there is only a terrible Venus-like fate to this planet. And that is an important issue, no doubt. Continue reading...

Forcing. This Has To Stop.

Most, if not all, set theorists at one point or another were asked by a fellow mathematician to explain how forcing works. And many chose to give as an opening analogy field extensions. You can talk about how the construction of an algebraic closure is a bit similar, since the generic filter is a bit like the maximal ideal you use to make this construction; or you can talk about adding a transcendental number and the things that change as you add it.

But both these analogies would be wrong. They only take you so far, and not further. And if you wish to give a proper explanation to your listener, there will be no escape from the eventual logic and set theory of it all. I stopped, or at least I'm doing my best, using these analogies. I do, however, use the analogy of "How many roots does $x^{42}-2$ has?" as an example for everyday independence (none in $\mathbb Q$, two in $\mathbb R$ and many in $\mathbb C$). But this is to motivate a different part of the explanation: the use of models of set theory (e.g. "How can you add a real number??", well how can you add a root to a polynomial?) and the fact that we don't consider the universe per se. Of course, in a model of $\ZFC$ we can always construct the rest of mathematics internally, but this is not the issue now. Just like we have a model of one theory, we can have a model for another. Continue reading...

John Oliver (and his team of writers, I suppose) makes a particularly sharp point about the role of the media in the debate about climate changes.

Downward Löwenheim-Skolem Theorems and Choice Principles

I have posted a new note on the Papers page.

It's a short little proof that the classic downward Löwenheim-Skolem theorem is equivalent to $\DC$, and that for a well-ordered $\kappa$, the downward Löwenheim-Skolem asserting the existence of models of cardinality $\leq\kappa$ is in fact equivalent to the conjunction of $\DC$ and $\AC_\kappa$. Continue reading...

...And we're back!

Okay, I took the time to make some changes to my homepage.

Clearly, the theme is different now. I also changed the content of the Papers page. I removed the abstracts (for some reason I thought this is going to be a cool thing to have, but with time it grew to annoy me greatly). I will definitely post a few things there in the coming time, some notes and eventually some nice papers -- I hope! Continue reading...