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Comment by ronaldx | Hacker News Reader

As best as I understand, they've proven the following:

If 3x = 3y, then x = y.

(and that's all)

They claim: "In this paper we show that it is possible to divide by three,"

and they've done this in the sense that dividing something by 3 has only one unique solution - i.e. dividing by 3 will never give you an ambiguous result.

This may seem obvious to anyone who is past kindergarten. But, the point of the paper is that they've done this from the ground up: starting with the fundamental Zermelo–Fraenkel axioms (and without using the modestly-controversial axiom of choice) https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

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So the "kindergarten obvious" statement is:

"For all natural numbers x and y, if 3x = 3y, then x = y"

An equally trivial but rather more abstract way to phrase this is in the category of finite sets:

"For all finite sets x and y, whenever there's a bijection between the product x cross 3 and the product y cross 3, then there's a bijection between x and y". (Here "3" represents any set of cardinality 3, but typically chosen to be the set {0,1,2} where 0 = {}, 1 = {0}, 2 = {0,1}).

What's (apparently!) not so trivial, is when you remove the "finite" restriction from the above, and you want to show it holds for all sets, including those of various infinite cardinalities. Actually constructing the bijection seems like it would require an infinite number of arbitrary choices, which is why it's impressive they don't rely on the axiom of choice.

I'm reminded of a professor who told us that we shouldn't distract ourselves with the concept of "infinity". I'm pretty sure he would have felt the same way about kindergartners.

Since I made some genuine effort to answer the parent query, I would be curious to hear why I'm attracting significant downvotes?

I don't know -- I didn't downvote you, but you might want to edit to indicate that x and y are not necessarily finite sets. (For finite sets, the result is trivial.)

Thanks. I think mjw's comment, and yours, now correctly cover this :)

> If 3x = 3y, then x = y. (and that's all)

No, that's not quite all. They prove it 1) even if x and y are infinite and 2) without resorting to the axiom of choice. It is those two things that make it interesting.

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