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

## In praise of Replacement

I have often seen people complain about Replacement axioms. For example, this MathOverflow question, or this one, or that one, and also this one. This technical-looking schema of axioms state that if $$\varphi$$ defines a function on a set $$x$$, then the image of $$x$$ under that function is a set. And this axiom schema is a powerhouse! It is one of the three component that give $$\ZF$$ its power (the others being power set and infinity, of course).

You'd think that people in category theory would like it, from a foundational point of view, it literally tells you that functions exists if you can define them! And category theory is all about the functions (yes, I know it's not, but I'm trying to make a point).

## Definable Models Without Choice

Suppose that a parameter formula defines an inner model. Does that inner model satisfy choice?

Well, obviously, if choice failed then the answer is no, just by taking $$x=x$$. But what if we remove that option. Namely, if the inner model is not the entire universe, then choice holds.

## Some thoughts about teaching introductory courses in set theory

Dianna Crown, the physics woman on YouTube, has posted a video where she is interviewed by her editor about why and how she found herself majoring in physics in MIT.

Here is the video:

## The transitive multiverse

There are many discussions on the multiverse of set theory generated by a model. The generic multiverse is given by taking all the generic extensions and grounds of some countable transitive model.

Hamkins' multiverse is essentially taking a very ill-founded model and closing it to forcing extensions, thus obtaining a multiverse which is more of a philosophical justification, for example every model is a countable model in another one, and every model is ill-founded by the view of another model. The problem with this multiverse is that if we remove the requirement for genericity, then everything else can be satisfied by the same model. Namely, $$\{(M,E)\}$$ would be an entire multiverse. That's quite silly. Moreover, we sort of give up on a concrete notion of natural numbers that way, and this seems a bit... off putting.

## Mathematical philosophy on YouTube!

If you follow my blog, you probably know that I am a big fan of Michael Stevens from the VSauce channel, who in the recent year or so released several very good videos about mathematics, and about infinity in particular. Not being a trained mathematician, Michael is doing an incredible task.

Non-mathematicians often tend to be Platonists "by default", so they will assume that every question has an answer and sometimes it's just that we don't know that answer. But it's out there. It's a fine approach, but it can somewhat fly in the face of independence if you are not trained to think about the difference between true and provable.

## Some thoughts about "automated theorem searching"

Let me begin by giving a spoiler warning. If you haven't watched "The Prisoner" you might be spoiled about one of the episodes. Not that matters, for a show from nearly 40 years ago, but you should definitely watch it. It is a wonderful show. And even if you haven't watched it, it's just one episode, not the whole show. So you can keep on reading.

So, I'm fashionably late to the party (with some good excuse, see my previous post), but after the recent 200 terabytes proof for the coloring of Pythagorean triples, the same old questions are raised about whether or not at some point computers will be better than us in finding new theorems, and proving them too.

## Quick update from Norwich

It's been a while, quite a while, since I last posted anything. Even a blurb.

I'm visiting David Asperó in Norwich at the moment, and on Sunday, the 12th, I will return home. It seems that the pattern is that you work most of the day, then head for a few drinks and dinner. Mathematics is eligible for the first two beers, philosophy of mathematics for the next two, and mathematical education for the fifth beer. Then it's probably a good idea to stop. Also it is usually last call, so you kinda have to stop.

## What I realized recently

I recently learned that diamonds are cut and polished with the dust of other diamonds. And I recently realized that success is cut and polished with the dust of failures.

In particular a successful mathematical idea is polished with the dust of the many failed ideas that preceded it.

## The Five WH's of Set Theory

I was asked to write a short introduction to set theory for the European Set Theory Society website. I attempted to give a short answer to what is set theory, why study it, when and how to study it and where to find resources.

You can find the article on the ESTS' website "Resources" page, or in the Papers section of my website.

## Don't worry about it

In a recent Math.SE question about the foundations of category theory without set theory, someone made a claim that $$\ZF$$ makes it hard to learn mathematics, because in $$\ZF$$ the questions "is $$\RR\subseteq\pi$$?" and "is $$\RR\in\pi$$?" can be phrased. They continued to argue that there are questions like whether or not hom-sets are disjoint or not, which are hard to explain to people who are "drunk on ZF's kool-aid".

So I raised a question in the comment, and got replies from two other people who kept repeating the age old silly arguments of what are the elements of $$\RR\times\RR$$ or what are these or that elements. And supposedly the correct pedagogical answer is "It does not matter what are the elements of $$\RR\times\RR$$." With that I strongly agree, and when I taught my students about ordered pairs on the very first class of the semester, I made it very clear that there are other ways to define ordered pairs and that we only do that because we want to show that there is at least one way in which ordered pairs can be realized as sets; but ultimately we couldn't care less about what way they encode ordered pairs into sets, as long it is a "legal" way.

## Name that number

In the best TV show ever produced, Patrick McGoohan plays the mysterious No. Six. He lives in The Village, where former spies are held. The people there are essentially captive, and they all have numbers instead of names. But he is not a number! He is a free man!

We find a similar concept in Zelda's poem "Every man has a name" (לכל איש יש שם), which in Israel is closely associated with the Holocaust and with assigning numbers to people. But alas, we are all numbers in some database. Our ID numbers, employer number, the index under which you appear in the database. You are your phone number, and your bank account number. You are the aggregation of all these numbers. And more.

## Is the Continuum Hypothesis a definite problem?

I am not a Platonist.

In general, while I do find it entertaining to think about god, afterlife, or a concrete mathematical universe, I find more comfort in the uncertainty of existence than I do in the likelihood that my belief is wrong, or in the terrifying conviction that comes along with believing in something (and everyone else is wrong).

## The Torture of Mathematical Research

In a manner more befitting to Edgar Allan Poe, Mathematics is a cruel and unforgiving mistress.

Mathematics will often dangle in front of you some ideas, and you will work them out, to find a mistake. Then you will go back to the beginning, find new ideas that she had in store, work those out and proceed only to find a mistake much later. Then you go back to the beginning, and you find yet another minor idea that was missing, and now when everything works you continue. But then you find another gap, and you have to go back to the beginning and hope to find yet another idea. And don't get me started on those ideas that you find not to work during all these searches.

## Existentialism II, like Colonel Kurtz

Last night I posted a strange story about a gecko and a moth.

It occurred to me today that this is a very Kurtzian story, if we take the Brando interpretation of Mistah Kurtz (he dead) in Apocalypse Now! (the Redux version is one of my favorite movies, I guess). In the movie Harrison Ford plays a tape where Kurtz is describing a snail crawling along the straight edge of a razor, crawling slithering, this is his dream, this is his nightmare.

## Existentialism

Spring has begun in Israel.

Yesterday was the first day where you could say that the weather is characteristically spring; and today (as well tomorrow) we are expected for a daytime heatwave and a nighttime cold weather (e.g. Beer-Sheva is expecting a whopping 31 degrees centigrade during the day, and 13 during the night).

## 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.

## 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.

## The cardinal trichotomy: finite, countable, and uncoutnable.

There is a special trichotomy for cardinality of sets. Sets are either finite, or countably infinite, or uncountable. It's an interesting distinction, and it has a very deep root -- at least in my perspective -- in the role of first-order logic.

Finite objects can be characterized in full using first-order logic. The fact that you can write down how many elements a set have, is a huge thing. For example, every finite structure of a first-order logic language has a categorical axiomatization. If the language is finite, then the axiomatization is finite as well.

## Choice Principles: What are they?

What does the phrase "$$\varphi$$ is a choice principle" mean? This is something that I have spent quite a lot of my time thinking about. Directly and indirectly. What are choice principles as we know them? And who gets to decide?

For a set theorist, at least a "classical" set theorist (working within the confines of $$\ZF$$ and its extensions to $$\ZFC$$ and so on), a choice principle can aptly be defined as "Sentence $$\varphi$$ in the language of set theory which is provable from $$\ZFC$$ but independent from $$\ZF$$". Indeed that is how I think of choice principles, and how I referred to them in my masters thesis (albeit I prefaced that definition by pointing out its naivety).

## The Philosophy of Cardinality: Pathologies or not?

What are numbers? For the layman numbers are those things we use for counting and measuring. The complex numbers are on the edge of being numbers, but that's only because they are taught in high-schools and many people still consider them imaginary (despite them having some reasonably applicative uses).

But a mathematician knows that a number is basically a notion which represents a quantity. We have so many numbers that I don't even know where to begin if I wanted to list them. Luckily most of the readers (I suppose) are mathematicians and so I don't have to.
