
Cantor's Diagonal Proof - msvan
http://www.mathpages.com/home/kmath371.htm
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tambourine_man
I was very relifed to discover later in life that I wasn't alone in
disagreeing with Cantor. That, along with a few other proofs and axioms, were
a major source of angst for me as a kid in school.

I find Wittgenstein's take on it very interesting.

[http://plato.stanford.edu/entries/wittgenstein-
mathematics/](http://plato.stanford.edu/entries/wittgenstein-mathematics/)

I wish some teacher had told me that other maths are possible if we agree with
different rules, even though schools are always going to teach and require the
status quo.

~~~
ufo
How do you define real numbers without uncountable sets though? I'm not sure
most people are willing to give up the real numbers just because they are a
bit unintuitive.

~~~
VladRussian2
real numbers are a closure of the set of rational numbers under the limit
operation. It just happens that the closure is uncountable.

It is pretty standard device in mathematics - having a set and an operation,
to produce and explore another set built as a closure of the first set under
the operation. Seems pretty intuitive to me :)

What is really unintuitive, puzzling is the starting point of it all - natural
numbers... 1, 2, 3... why there is a one stone, a one star, ... why this
discretization?

~~~
eru
You can start lower, if you want to.

{} == 0

{{}} == 1

...

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ColinDabritz
A key part of Turing's computability paper hinges on the diagonal argument.
It's really well covered in 'The Annotated Turing' by Charles Petzold.

In Cantor's version, we prove that numbers (real numbers?) are not enumerable
by creating a new number.

In Turing's paper, he enumerates all valid Turing machines in a similar way,
but crucially, we can't know if the new constructed machine is valid or not,
so the outcome is different.

It's quite possible I'm botching elements of these, but it was a beautiful
read.

~~~
thaumasiotes
> In Cantor's version, we prove that numbers (real numbers?) are not
> enumerable

in the standard diagonalization, you just prove that the reals in the interval
(0, 1) are uncountable; that proof is very simple to state and then you can
argue that (0, 1) is a subset of the reals so the real line must be
uncountable too.

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Steko
Intro to Real Analysis (or whatever your school calls it) -- where Cantor's
proof is normally covered -- is really a gem of a course and the first time
since 8th/9th grade geometry that students are clearly shown why math is the
queen of the sciences.

Unfortunately, in my experience, a lot of future high school math teachers
skipped it and not because it's hard but because the class before it
(intermediate calc) was a huge step up from high school calculus (yes they
should take intro to calc at college level but kids who do well in high school
calc always want to skip it). What ends up happening is you get a lot of high
school calc teachers who don't really understand calculus all that well and
can't answer questions and so they just end up teaching to the AP exam and the
vicious cycle reinforces itself.

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ketralnis
I read this for the first time in the excellent and surprisingly accessible
The Annotated Turing[0], which I can highly recommend. If you're vaguely
interested in things like proofs like these or about computability or just
Turing's and others' contributions and approaches, the book approaches these
things very well without presuming a deep pure mathematical background.

Seriously, read it.

[0] [http://www.amazon.com/Annotated-Turing-Through-Historic-
Comp...](http://www.amazon.com/Annotated-Turing-Through-Historic-
Computability/dp/0470229055/ref=sr_1_1?s=books&ie=UTF8&qid=1383018325&sr=1-1&keywords=the+annotated+turing)

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gtremper
One of my favorite proofs. It blew my mind when I understood it; it's so
simple in hindsight. I've tried to explain it to non-technical people and
haven't has as much luck conveying why it's so cool.

~~~
GuiA
The book "The Cat in Numberland" is a fantastic way to introduce this proof
(and a few other related mathematical concepts) to non-mathematicians and
young children.

It is however becoming harder and harder to find for a reasonable price :( I
paid $50 for mine, which is quite a bit for a children's book, but it is that
good.

[http://amzn.com/081262744X](http://amzn.com/081262744X)

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ttiitg
"The digits of every rational number repeat after some finite number of
digits, so the "period" of every rational number is finite. However, there is
no upper bound on the period of rational numbers, i.e., the periods are all
finite, but there is no largest period. Thus, in a manner of speaking, the
least common multiple of this set of strictly finite things is infinite."

Got lost here, what is the LCM of this set; which set?

~~~
thaumasiotes
this set (the set of periods of decimal expansions of rationals) seems to be
oddly defined to no purpose.

Consider the set of positive integers. It has the same property described (in
fact, it has all the same properties, since it's the same set). Why go to all
the trouble of defining Z+ as "the set of periods of decimal expansions of
rationals", which is harder to parse?

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bumbledraven
Here's a proof of Cantor's Theorem in Metamath, which is one of the simplest
formal systems in the world:
[http://us.metamath.org/mpegif/ruc.html](http://us.metamath.org/mpegif/ruc.html)

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thenerdfiles
[https://gomockingbird.com/mockingbird/#xl6a68x](https://gomockingbird.com/mockingbird/#xl6a68x)

Here I use the Diagonal Proof to talk about Observation Statements (or
Observa) / Mental States (of Observers) such that Diagonal Statements (DiaSt),
once normalized for all Observers, can be measured along arbitrary
distribution of duration ranges (T) of a non-local universe. Given this, we
can predict the upper/lower bound of periodic collapse of quantum systems (our
Model).

I've also seen Cantor's Proof applied as a metaphor for the analysis of Web
Artefacts: Hypermedia Types (X axis) / Web Components (Y axis). (I see "soft
objects" as a species of Web Artefact.)

There are many interesting applications. For me, personally, I think Cantor's
Proof gives us a means of assuming the periodic collapse of quantum systems
w/r/t quantum observers (it is a myth that "your" mind and "my" mind are
discrete entities) such that quantum field theory can be salvaged.

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kriro
I always liked the explanation of the proof in Gödel, Escher, Bach. Very easy
to follow.

