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An introduction to Georg Cantor’s transfinite paradise (medium.com)
98 points by pleko 4 months ago | hide | past | web | favorite | 40 comments



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

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


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...


Deep!


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)


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


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.


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.


What is category in this sense?


a baire category is an old-timey synonym for a baire space. It's related to topology & analysis, not algebra (like category theory). In the modern theory we can make an analogy (but it is just an analogy!) between 'categories' of (complete non-empty) metric spaces on the one hand, and measure spaces (of positive measure) on the other, as follows:

  metric spaces:    measure spaces:

  first category    zero measure
  second category   positive measure
  residual          full measure
  baire             measurable


A terminology note, I think "first category"/"second category"/"residual" have largely been replaced by the clearer "meagre"/"non-meagre"/"comeagre". Or at least, I hope they've largely been so replaced, because geez the old terminology is a pain. :P


Clearer?!?


Yes. "Meagre" means small. Which is what a first-category set is, but the name "first category" doesn't tell you that (how is one supposed to remember which of first and second category is which?). And then after that, you don't need to introduce any more words. If you mean to say something is not meagre, you just say it's not meagre, same as you would with any concept. If you mean to say it's complement is meagre, you just say it's comeagre, same as with anything.

The names "first category" and "second category" tell you nothing, not even making it obvious which is which -- and moreover, they make it sound like the two are both equally useful sides of a distinction, like finite and infinite, while in reality one of these is actually directly useful and the other is just everything that isn't that (which is how most distinctions in mathematics go; you wouldn't make up a separate new word for "not compact", for instance).

And then "residual" makes it sound like there's some new thing going on, when of course it's really just being co-first-category. But co-first-category's quite the mouthful, isn't it...?


I'd say so. "non-meagre" = not meagre = positive, and "comeager" = complement of meagre = full. The only thing you have to remember is what "meagre" means.


i remember in undergrad my analysis prof saying there were two types of analysis: hard analysis (sans baire category theorem) and soft analysis (using baire category theorem). i wonder if what he actually meant was in baire spaces and not in baire spaces.


I suspect your professor was referring to 'qualitative' vs 'quantitative' approaches when referring to soft vs hard analysis. The baire category theorem is intimately related to these to approaches, and leads to three theorems (1) the uniform boundedness principle, (2) the open mapping theorem, and (3) the closed graph theorem, that establish equivalences between the qualitative and quantitative theories.

According to Tao's textbook, in practice these are not used much directly, instead one starts with quantitative bounds, then derives the qualitative corollaries. The baire category theorem tells us why those there theorems are true, and thus tells us why we can take that step from quantitative to qualitative. I think this is what your professor meant.


can you give an example of a qualitative corollary? do you mean a characterization of a space or something like that?


Referring to properties of linear operators. Examples are finiteness, surjectivity, non-negativity, monotonicity, min/max principles, contraction properties, etc.


"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?!


...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.


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.


I see now. Thank you.


Infinitesimals forever! :-)


I'm not intuitively convinced that this proof is sound because I feel like is ignoring the concept of "stretching" when mapping a fixed range like (0,1) to a larger or smaller set. EG, I don't care that you can map points from (0,1) to (0,2) one to one, because I know that each element in (0,2) is forced to cover twice as much space (regardless of taking the limit so that the cover goes to zero, this cover ratio is always 2:1 ). It smells fishy, like Cantor invented a way of comparing set sizes that intentionally drops units of cover so that naturally distinct elements fall into the same equivalence class. Most arguments occur downstream of this choice, so debates are imprecise and people speak past each others points.

Unfortunately if you object to Cantor's ideas online, you will be flooded with people who will call you a moron and recite whatever garbage explanation they received as a child, instead of offering some thoughtful counterpoint. At the end of the day I think Cantor's ideas are a mathematical tool that the mainstream has run with, but isn't actually the most useful tool.

(Monster) Curves which have continuous dimensions should be the basis for creating a infinite set system with consistent utility. (EG, there are curves which are very "spiky" that they break free of being 1 dimensional but, are not quite an area (dimension 2) -- they could be perhaps 1.6 dimensional, IE- they _sort-of_ have an area. ) With Cantor-tinted glasses a normal 1 dimensional line would have the same cardinality of points as a 1.6 dimensional one, so some richness is lost.


I don’t think anyone should call you a moron. I do think you should consider that if professional mathematicians don’t have a problem with a proof and you do then it is most likely an indication of a lack of understanding on your part.

I don’t understand your objection because I’m not sure what exactly you mean by

...each element in (0, 2) is forced to cover twice as much space...

I will point out that saying that (0, 2) is a larger set than (0, 1) is wrong without specifying how you are using the term size. The size of the sets, in the context of set theory, are the same. However, in terms of measure theory they are not the same. The notion of size depends on what exactly is being measured. In one context (0, 2) is larger than (0, 1) but not in another context.

Unfortunately the term “size” has a connotation to most people and in set theory that term is not the connotation that most people have. A person will naturally think that an interval of length 2 is bigger than one of length 1. The word size doesn’t have any inherent meaning. It’s jut a word. It means what people agree to have it mean.

In set theory the accepted meaning of size corresponds to mappings between sets. The size of the set {0, 1, 2, 3} is less than the size of the set {0, 1, 2, 3, 4} because I can map the first set injectively into the second set but not vice versa. This is an intuitive sense of size when dealing with finite collections of objects. Intuition can break down when abstracting to infinite collections of objects.


Thanks for your helpful reply, but I am still having an issue understanding part of the idea of mapping sets of points.

Suppose that there was an arbitrary rule that points can only exist at integer coordinates, then (0,2] would have 2 elements and (0,1] would have one element. You would not be able to map 1 to one.

Now, suppose that points are allowed to exist at coordinates m*(1/n) where n=2. (EG, 0, .5, 1, 1.5 ...) So now the first set would have 4 elements, and the second would have 2. They would still not map one to one.

Now imagine that you can pick an n arbitrarily large to represent any rational number. I still don't see how you can map the sets one to one.

Is there a reason that you can map them one to one, or is the set theory that we use specifically constructed/designed to afford this property?


I think the problem comes from the mapping you are choosing. You can construct a map that between (0, 2) and (0, 1) that is onto but not one-to-one. But to say that (0, 2) and (0, 1) have the same size as sets (cardinality) means that there exists a bijective map between the two sets. You just haven’t found out what that mapping is. The mapping you are thinking of is not a bijection so from your perspective you are not seeing the sets as being the same size.

Consider this. The set of non-negative integers is {0, 1, 2, 3, 4,....} and the set of even non-negative integers is {0, 2, 4, 6, ...}. One might think the first set is twice as large as the second set. And under a natural way of looking at it this appears to be true. But consider this mapping:

0 -> 0

1 -> 2(1) = 2

2 -> 2(2) = 4

3 -> 2(3) = 6

.

.

.

Under this map I’ve bijectively mapped the set of all non-negative integers to the set of non-negative even integers. This is how intuition can fail when dealing with infinite sets. Under one way of looking at things one set appears to be larger than the other. But the definition of cardinality (size of the set) deals with whether or not a bijection exists and does not care that other mappings might give the appearance of one set being bigger than another set.

All infinite sets have this property: They can be put in bijecitve correspondance with a proper subset. This can not happen with finite sets.

Here’s a bijection from [0, 1] to [0, 2]. f(x) = 2x.


I mean, it's cleaner if you do it not for real numbers but rather for sets of whole numbers (which are of course in bijection with the reals). Then you can offload all such fiddliness to that one bijection, rather than dealing with it repeatedly in different contexts.


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

https://www.youtube.com/watch?v=SrU9YDoXE88


Thanks for all the upvotes!


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?


Beautifully written article.


"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.


Transfinite does not mean infinite. See: https://en.wikipedia.org/wiki/Transfinite_induction


According to the article:

"In his 1883 paper entitled Grundlagen einer allgemeinen Mannigfaltigkeitslehre (“Foundations of a General Theory of Manifolds”) he introduced a distinction between two infinities, the transfinite and the absolute:

"Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.

"The absolute infinite Ω, also introduced by Cantor, can be thought as a number which is bigger than any conceivable or inconceivable quantity, either finite or transfinite."


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


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.)


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


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.


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.


I totally agree with your first paragraph. If you or anyone can suggest an accurate, neutral title that has this property, we'd happily change it again. "Introduction to the Continuum Hypothesis" isn't quite accurate.




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