# The desire to be puzzled

Here is a very interesting post on possible alternatives to Cantor’s Transfinite Numbers Theory! Was Cantor’s notion of infinite really “unavoidable”?
The topic is particularly interesting when considered within the general framework of mathematical development. How does mathematics evolve? What “forces” mathematics to take some roads instead of other roads? And how do these choices influence the way in which we model and understand the natural world?

A prominent professor in the philosophy of mathematics once told me that the key to writing an attractive philosophy paper is to present the reader with a puzzle. “Give me a puzzle, and I’ll be interested”, he said. As I was surrounded by mathematicians and philosophers of mathematics which were steadily exchanging puzzles, I had no doubt that he was right: mathematicians and philosophers of mathematics like puzzles. But then, mightn’t it be the case that this fondness of puzzles influences much more than just our judgment of a philosophy paper (and our conversations over dinner)? Here’s a crazy idea – or maybe not so crazy – does our desire to be puzzled affect our judgement of a certain foundational mathematical theory?

The foundational mathematical theory which I have in mind is, of course, Cantor’s transfinite set theory. Given its general acceptance nowadays, it is easy to forget that in order to generalize arithmetic from the finite to…

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The argument is quite simple: if numbers were sets, we should be able to find a unique progression of sets with which numbers can be identified. But this is apparently impossible: there is a lot of ω-series that can serve as well for the aim. For example, we can adopt von Neumann’s series, and say that $0=\emptyset, 1=\{\emptyset\}, 2=\{\emptyset,\{\emptyset\}\}$, and so on, where the successor function $S$ is defined by $S(x)=x \cup \{x\}$. Or we can adopt Zermelo’s series, and say that $0=\emptyset, 1=\{\emptyset\}, 2=\{\{\emptyset\}\}$, and so on, where the successor function $S$ is defined by $S(x)=\{x\}$. Now, the problem is: is $3=\{\{\{\emptyset\}\}\}$ or is $3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$? Benacerraf presents then the example of two children, Ernie and John. The first learned that von Neumann’s ordinals are the natural numbers, while the latter that Zermelo’s ordinals are the natural numbers. Now, they will be easily able to learn arithmetic set theoretically via the above constructions, and they will agree on any arithmetical theorem, except that for Ernie it is true that $3 \in 17$, while for John it is false!