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.

There is also Väänänen’s multiverse, which is more abstractly defined, and I cannot for the life of me recall its definition and its details.

Some time ago Ur Ya’ar gave a seminar talk about Hamkins’ multiverse in the logic seminar in Jerusalem. It was interesting, and afterwards Yair Hayut and myself talked with Ur about these multiverses. One idea that came up, and I don’t think that I ever ran into it, is sort of a combination between the generic multiverse and Hamkins’ multiverse. Consider the following axiom “Every real is an element of a transitive model”. Now look at $\cal M$, the set of all the countable transitive models, we get the following axioms are satisfied by $\cal M$:

  1. If $M\in\cal M$, then every generic extension and every ground of $M$ is also in $\cal M$.
  2. If $M\in\cal M$, then every inner model of $M$ is also in $\cal M$.
  3. If $M\in\cal M$, then there is some $N\in\cal M$ such that $M\in N$ and $N\models M\text{ is countable}$.
  4. For all $M,N\in\cal M$, $L^M$ and $L^N$ are comparable.

So what do we have here? We have a multiverse of sets, it is closed under generic extensions and grounds, and it is even closed under definable inner models. It also has the property that we can always find bigger models that think a given model is countable.

Now, I have no idea what useful things can come out of this multiverse. And I would imagine that one should first refine this notion a bit more before it becomes actually useful for something. But nonetheless, it seems like an interesting interpretation of the whole notion of multiverse.

Strong coloring

I am sitting in the 6th European Set Theory Conference in Budapest, and watching all these wonderful talks, and many of them use colors for emphasis of some things. But yesterday one of the talks was using “too many colors”, enough to make me make a comment at the end of the talk after all the questions were answered. Since I received some positive feedback from other people here, I decided to write about it on my blog, if only to raise some awareness of the topic.

There is a nontrivial percentage of the population which have some sort of color vision deficiency. Myself included. Statistically, I believe, if you have 20 male participants, then one of them is likely to have some sort of color vision issues. Add this to the fairly imperfect color fidelity of most projectors, and you get something that can be problematic.

Now, I’m not saying “don’t use any colors”. Not at all. Just keep in mind that some people might have problems with your choice of colors. Using too many colors can be distracting, and one of the slides in the said talk had black text almost on par with the rest of the colored text. This is far from ideal. But since color deficiency can vary from one to another, let me only give an account of my own personal experience. I cannot do anything more, after all.

I have a mild red-green issue. But this means also that yellow and bright green, or light orange, all mix together sometimes; and darker greens can be red or brown (which themselves are often mixed); and blues can mix with purple, and sometimes with pink as well. One other effect of color deficiency is that you are more sensitive to brightness and darkness (the eye compensates the damaged cones by having better rods, so your night vision gets somewhat better, for example).

So when you have a slide with some pink/purple and green/yellow/orange and some blue and some red and some black, my brain will not read the text. My brain will try to make sense of the colors. Not to mention the terrible eye strain coming from the brighter colors (here the quality of the viewing media is important, I’m sure that I’d be fine watching the same slides on a proper computer monitor). There were slides that I had to turn my eyes away from the talk. Yes, it was pretty bad.

What can you do about it? Don’t use colors when you don’t have to. Use boldface or italics for emphasis when possible, or different font family entirely. If you want to use colors, using them sparingly, and try to avoid relatively close colors together and certainly try to avoid brighter colors like light green or yellow. If you know a color blinded person, you can maybe ask them to give some critique on your choice of colors.

Some people commented to me after my remark that they prefer the colors, and they are helpful. I understand that. Again, the point is not to get people to use colors. Just… to use them intelligently. Colors are like spices. I’m not trying to get you to cook without spices, but you’re not going to serve a dish entirely made of cinnamon and cumin.

In the name of all color vision deficient people, thanks in advance for your consideration!