
Division By Three (2006) - frozenport
http://arxiv.org/abs/math/0605779
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transfire
Could someone give a layman's overview of this paper?

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ronaldx
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...](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)

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mjw
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.

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jessaustin
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.

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frozenport
"Their failure to give anything of a proof must have frustrated Sierpinski,
for it appears that twenty years later he still did not know how to divide by
three."

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gweinberg
Is there something special about 3? I would think the result would be true for
any positive finite n.

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gjm11
The authors say (1) that division by 2 is much easier (though not trivial) and
(2) that their methods generalize to enable you to divide by any other odd
number (and hence, together with division by 2, to divide by any positive
integer).

So what's special about 3 is that it's the smallest tricky case and that once
you've done it you've effectively done everything.

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sfermigier
I find it disturbing that, as stated in the first footnote, John Conway can be
listed as an author of the paper, against his approval (or despite his non-
approval).

What if someone gets hurt because of this paper and Mr Conway's responsibility
is engaged ?

