I just posted a note on PhilPapers, which is also forthcoming in Philosophy of Science. (It is a bit weird for a completely mathematical paper to show up in Philosophy of Science, but it is what it is…)
The main feature of this note is that it describes a question that I could not solve. But this question is so simple that it can be explained to any mathematics undergrad or even high school student. So here we go…
Cantor tells us that infinite sets have different sizes.
Click for quick review of Cantor…
According to Cantor, the size of set \(A\) being equal to the size of set \(B\) means there is a perfect correspondence between them. For example, we have \(0\mapsto 0, 1\mapsto 2, 2\mapsto 4, \dots, n\mapsto 2n, \dots\), so the set of natural numbers \(\{0, 1, 2, 3, \dots\}\) and the set of even numbers \(\{0, 2, 4, \dots\}\) have the same size. Sets that have the same size as the natural numbers are called “countable”, and they are the smallest among the infinite sets. On the other hand, the set of real numbers is larger than countable sets, since Cantor discovered that for any countable list of real numbers we can always find yet another real number that is not in the list.
Cantor’s theory has a interesting feature: although the even numbers are only “half of” the natural numbers, in Cantor’s theory these sets have the same size!
Consequently, the possible number of sizes is quite small. For example, any infinite subset of a set of size \(\aleph_7\) (the 8th level of infinity, with the countable sets at the 1st level \(\aleph_0\)) must be at one of these 8 levels — although there are \(2^{\aleph_7}\)-many subsets, a huge number in comparison to 8! In other words, we are classifying these \(2^{\aleph_7}\)-many subsets into merely 8 baskets.
What if we want to say that the set of even numbers has a smaller size than the set of natural numbers? We would need an alternative definition of size. The part-whole principle says if set \(A\) is strictly contained in set \(B\), then the size of \(A\) is strictly less than the size of \(B\).
The question I could not solve is the following: suppose we are given an infinite set \(\kappa\), and we assign a size for its subsets, while respecting the part-whole principle. That is, we are classifying \(2^\kappa\)-many subsets into a number of baskets, according to their size. What is the minimum number of baskets that we need?
Clearly we would need many more baskets than Cantor (e.g. the natural numbers and the even numbers were in the same basket, now we need two different ones), and this number will grow as \(\kappa\) grows. But how many?
In my note I determined that this number is at least \(\text{ded } \kappa\) — a quantity associated with \(\kappa\) that is greater than \(\kappa\), an improvement over previous results.
The quantity \(\text{ded } \kappa\) is of independent interest, since it relates to another bizarre feature of the infinite. Now let me briefly tell you what it is.
Think of a line segment, a rope, a banana, or what have you. If you cut it once in the middle, you get two segments. If you cut it 2 times, you get 3 segments. If you cut it \(n\)-times, you get \(n+1\)-many segments. Sounds quite trivial, right?
However, the situation is a bit different when you can cut infinitely many times. Think of your line segment as the interval \([0, 1]\) of real numbers. Now let us make cuts at all the rational numbers. How many segments do we get? Observe that any two irrational numbers belong to different segments, since between them there is a rational number where we made a cut, by the density of rationals. From Cantor, we know that there are countably many rational numbers but uncountably many irrational numbers. So this is indeed a very bizarre situation: with countably many cuts, we can produce uncountably many pieces! This forms a sharp discontinuity with the finite case, where the number of pieces \(n+1\) is effectively the same as the number of cuts \(n\).
One mathematician who identified this phenomenon was Richard Dedekind, who actually defined real numbers as the collection of cuts that you can make on rational numbers. What if we make even more than countably many cuts?
Consider an infinite number \(\kappa\) and let us make \(\kappa\)-many cuts. The maximal number of pieces you can get is exactly \(\text{ded }\kappa\), named after Dedekind. Just as we observed, you can cut countably many times and get uncountably many pieces, it is generally true that \(\text{ded }\kappa>\kappa\).
Cantor and Dedekind, generated by ChatGPT Image 2
Finally let me tell you why I can’t solve my problem.
We have \(2^\kappa\)-many sets to be classified, and it is known that it is possible to classify them into \(2^\kappa\)-many baskets while respecting the part-whole principle. Now that I showed one needs at least \(\text{ded } \kappa\)-many baskets, if \(\text{ded }\kappa=2^\kappa\), the question would be solved. But the problem is, while \(\text{ded }\kappa\le 2^\kappa\), whether they are equal for all \(\kappa\) is independent of the fundamental assumptions of mathematics, known as \(\mathsf{ZFC}\). In other words, there are some mathematical universes where \(\text{ded }\kappa< 2^\kappa\) at some \(\kappa\). Can we make the assignment with less than \(2^\kappa\)-many baskets in these situations?
I tried to settle this question for a bit but it appears beyond me. I hope it will be answered one day by some mathematician or perhaps, some AI! I can see several possibilities of how this turns out…
An argument (perhaps relatively simple but hitherto evades us) shows that we always need at least \(2^\kappa\)-many baskets.
A \(\mathsf{ZFC}\) construction (perhaps somewhat involved) shows that it is always possible to use merely \(\text{ded }\kappa\)-many baskets.
A forcing construction shows that there is a \(\mathsf{ZFC}\)-universe where we can use less than \(2^\kappa\)-many baskets.
My first academic paper “Are nonmeasurable sets significant for epistemology?” is forthcoming in Synthese. A preprint is available here.
While this paper may appear somewhat technical on the surface, its philosophical message is in fact extremely simple. Once I was asked by an anthropologist friend to explain what is going on in this paper in a bar (not the quietest place in the world). Here’s what I said:
In scientific investigations, it is often helpful to distinguish the substantial part of the phenomenon in question from the derivative features introduced by one’s instruments. On my view in this paper, some mathematical constructs known as “nonmeasurable sets” fall into the second kind when we are interested in probability and statistics, and therefore should not be considered too important.
The highlight of the paper is the following quote by Henri Poincaré, one of my academic heroes:
Le géomètre cherche toujours plus ou moins à se représenter les figures qu’il étudie, mais ses représentations ne sont pour lui que des instruments ; il fait de l’a géométrie avec de l’étendue comme il en fait avec de la craie; aussi doit-on prendre garde d’attacher trop d’importance à des accidents qui n’en ont souvent pas plus que la blancheur de la craie.
La science et l’hypothèse, 1917
Translated as: “the geometer always more or less seeks to represent the figures he studies, but these representations are merely instruments for him. He does geometry with the notion of extension just like he uses the chalk, moreover we should be cautious not to attach too much importance to accidental features which are often nothing more than the whiteness of the chalk.”
Poincaré and the three body problem (a very weird version thereof), as generated by ChatGPT
Several things to note:
Unlike many other philosophical works, I think mine is amenable to refutation/falsification. For example, the position in this paper would be significantly challenged if one identifies a theorem fundamental for probability that is not available in \(\mathsf{ZF+DC}\).
One strong candidate for such refutation is the Hahn-Banach theorem, which is fundamental for functional analysis, as well as for constructing certain finitely additive probability measures. Whether this theorem plays any role in probability and statistics is debatable.
In Footnote 11 of the paper one can find a curious quote by Hermann Weyl in Das Kontinuum, which I find highly interesting but not very comprehensible, since I have trouble understanding what kind of mathematical phenomena he was commenting on. If any reader has any insights on this, please communicate to me.
So much for the synopsis, please read the paper if you are interested.
Caveat: this post has nothing to do with mice as in set theory…
Infinitely many mice, as generated by DALL“-E…
I ran into the following puzzle lately and it took me a while to come up with a solution:
Suppose there are countably many mice standing in the following sequence: Mouse 0, Mouse 1, Mouse 2, and so on. Each mouse is given a hat which is either red or blue. Each mouse can see the hat color of every mice standing after it, but not their own, nor anyone standing before. Now, each mouse have to guess their own color. The mice collectively win the game if all but finitely many of them guesses correctly. Find a winning strategy for the mice. (Hint: Axiom of Choice.)
If you haven’t seen this, think a bit on your own!
Ever since the release of ChatGPT, I got interested in observing how generative AI interacts with logical and mathematical tasks. So I want to see ChatGPT’s performance in solving this problem. I am using ChatGPT3.5, and here is what I get.
If you read til the end of our conversation, you already know the solution, which is rather transparent by hindsight — modulo the equivalence relation of “equality at all but finitely many points”, each mice get the same information, namely the equivalence class of the input sequence, and they win if they collectively output a sequence in the same class. Now, before the game they fix a representative for each class (by an application of the Axiom of Choice). After the hats are given, everyone observes that they are in some class \([r]\), where \(r\) is the common representative chosen beforehand. They then win the game by collectively playing \(r\), i.e. Mouse \(N\) plays the \(N\)th digit of \(r\).
