
An introduction to Georg Cantor’s transfinite paradise - pleko
https://medium.com/cantors-paradise/the-nature-of-infinity-and-beyond-a05c146df02c
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sn41
A very nice article. It covers the connection of Fourier series to the initial
attempts in Set Theory, something that is not often mentioned. As far as I
know, at least three major notions of "largeness" came out of the attempts to
quantify the allowed sets of discontinuities in Fourier (in general,
trigonometric) series:

1) uncountability (Cantor)

2) category (Baire)

3) Measure (Jordan/Borel/Lebesgue)

~~~
achillesheels
Can you please elaborate on the Cantor-Fourier connection more explicitly or
provide a helpful URL? Thank you.

~~~
sn41
It is detailed in the article itself. Cantor wanted to say what classes of
sets of discontinuities are allowed such that we can still represent a
function with a trigonometric series. Cantor's approach immediately led to
what are now called Cantor-Bendixson derivatives, and indirectly led to Cantor
developing the rudiments of modern (naive) set theory.

~~~
achillesheels
I see, I missed it while I was looking for Fourier, thanks.

I’m curious how this work is mathematically related to the complex unit
circle.

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dannykwells
This article doesn't talk much about the Cantor set, which is a pity given how
amazing it is. Maybe one of the most beautiful objects in all of mathematics:

[http://mathworld.wolfram.com/CantorSet.html](http://mathworld.wolfram.com/CantorSet.html)

A great, rare, example of a set with no length (0 measure) but an uncountable
number of points.

~~~
antidesitter
Another nice example is the _fat_ Cantor set [1] which (like the Cantor set)
is nowhere dense but (unlike the Cantor set) has _positive_ measure!

[1]
[https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9...](https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set)

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mcguire
" _Proof that |ℝ²| = |ℝ| It suffices to prove that the set of all pairs (x,y),
0 < x,y < 1 can be mapped bijectively onto (0,1]. Consider the pair (x,y) and
write x,y in their unique non-terminating decimal expansion as in the
following example:..._"

I hadn't seen this proof before. I assume the "unique non-terminating decimal
expansion" uses the 0.999... = 1.0... shenanigan due to "Since neither x nor y
exhibits only zeroes from a certain point on..."

Further, " _Reportedly, Kronecker disagreed fundamentally with the thrust of
Cantor’s work on set theory because, among other reasons, it asserted the
existence of sets satisfying certain properties without giving examples of
specific sets whose members satisfied these properties. Kronecker also only
admitted mathematical concepts if they could be constructed in a finite number
of steps from the natural numbers, which he took for a given._ " Kronecker was
a constructivist?!

~~~
skh
_...uses the 0.999... = 1.0... shenanigan..._

Why is this a shenanigan? Do you dispute that 0.9999....=1? One easy way to
see this is that there are no decimals between 0.9999..... and 1 and thus the
two numbers are the same. A property of the real number system is that between
any two distinct numbers is another distinct number.

Maybe I'm missing the intent of your statement.

~~~
joppy
The slightly surprising incorrect statement is “each real number has a unique
decimal expansion”, which is wrong since 1.00000... = 0.99999... — so when
using arguments like this, you need to be careful to always only pick one of
these.

~~~
skh
I see now. Thank you.

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jonahx
This is a terrific vsauce video on transfinite numbers, which assumes almost
no background:

[https://www.youtube.com/watch?v=SrU9YDoXE88](https://www.youtube.com/watch?v=SrU9YDoXE88)

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jorgenveisdal
Thanks for all the upvotes!

~~~
aidos
Thanks for the article! While I studied set theory, I’d never had a run
through of any of the history before. Like a fine constructive proof, your
article makes it much clearer how all the parts fit together into a logical
whole :-)

It’s depressing to hear how it ended for Cantor. Especially given how modest
he comes across in his letters.

One thing I’m confused about with the plane to line bijection - how can you be
sure you’ll eventually hit another 0? What about .11111...? There you can
never switch over to the other dimension and the whole of the .11111 line on
the plane will have the same representation, no?

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nickdrozd
"Transfinite" is a crappy word. It sounds cool, but it means just the same
thing as "infinite" (namely "not finite"). Cantor devised it because he wanted
a way to say "infinite" without offending certain theological sensibilities.

~~~
pickdenis
Transfinite does not mean infinite. See:
[https://en.wikipedia.org/wiki/Transfinite_induction](https://en.wikipedia.org/wiki/Transfinite_induction)

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mark_l_watson
Great read, I enjoyed learning the math and Cantor’s eventually sad life
story.

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HerrMonnezza
This is a biographical page about Cantor, rather than an article about the
Continuum Hypothesis proper. (Which is explained, as are many of Cantor's
results, as narration proceeds along his life and achievements as a
mathematician.)

~~~
dang
The submitted title was 'An Introduction to the Continuum Hypothesis'. We
changed it to the article subtitle which seems more informative than the
article title.

" _Please use the original title, unless it is misleading or linkbait; don 't
editorialize._"

[https://news.ycombinator.com/newsguidelines.html](https://news.ycombinator.com/newsguidelines.html)

~~~
tolmasky
It's funny because I was just thinking "wow, this is such a great way to do an
introduction to the continuum hypothesis". You get the same "introduction"
that Cantor did: you walk through his early successes and learning that lead
to ask these important questions. I remember wishing math was taught this way
more generally (especially in grade school), where these results are often
just thrown at you as if they came from the sky, whereas, at least for me,
they make so much more sense when you first learn about the failures and _why_
we arrived at these ideas. The only time that really stands out to me where
this is done is with integrals and doing rectangles and newton's method first.

Anyways, all this to say, I clicked the link with the original title and was
very happy, but I probably would not click the current link since I wouldn't
really know what it means or what to expect.

~~~
andrewflnr
Interestingly, physics is taught the opposite way. It is routinely taught in
chronological order of discovery, and while I can't exactly fault it,
sometimes I wish they would ease off with faithfully relating every false
start.

