To the readers of my blog, it should come as no surprise that I have a lot of appreciation to what Michael Stevens is doing in Vsauce. In the past Michael, who is not a mathematician, created an excellent video about the Banach-Tarski paradox, as well another one on supertasks. And now he tackled infinite cardinals and ordinals.
You can find the video here:
And if you’re familiar enough with the intricacies of large cardinals, at some point you’ll start seeing minor problems. These are very minor indeed (and I am willing to ignore them in favor of having a cohesive and clear expositional video, like it is right now), like saying that a cardinal that cannot be reached from below by applying replacement to addition and power sets is an inaccessible cardinal. This is not false, but it’s not accurate either. The ambiguity lies in “from below”, and if you accept this as “the universe already exists”, then this is true; but if you think about the universe as something we are creating one step at a time, then this is also true for worldly cardinals which are not inaccessible. To reach a singular worldly cardinal “from below” we first need to go above it, and only there we can find the witness for its singularity.
If this gives you a hint of Platonism, wait until you hear him discuss the continuum hypothesis being unsolved. The first mention of CH in the video is fairly innocent mentioning just that we don’t know. But the later references to it imply that “there is an answer”, and this is a very Platonistic approach to set theory. Which is fine, I guess. The final nail in this coffin was the last statement where Michael says that there are some promising leads to solving the continuum hypothesis. (That’s what I have to say on the topic of CH)
So when the video was over, I looked at the description, and saw that he had several discussions with Hugh Woodin about this. And all became clear, and a lot of the small remarks in the video started to fit together in a way that I couldn’t put my finger on it before.
But don’t get me wrong. The video itself is still wonderful, and it is still a very good exposition to the idea of ordinals and cardinals. And even large cardinals. And everything is augmented times $\aleph_2$ when you know that Michael is not a mathematician.
Kudos, Michael, for another good job!
(The one thing that I couldn’t look past, especially after being historically accurate with Cantor and Zermelo, is at the end where the famous hierarchy of large cardinals is shown, and Michael asks whether or not we’ll come up with large cardinals axioms which imply a contradiction: we have done that, numerous times before.)