Cantor Set Puns

I really like math history and in a lot math history books, I always see authors briefly mention a big shift in math during this time period and I would like to read more into it. It can be hard to find good math history books on specific situations though.

Hello, I’m a math first year student and recently learned about Cantor’s theorem regarding the comparison of cardinalities between infinite sets.

I have had a lot of difficulty though with rationalizing a good intuition about some of the deductions derived from it, specially ones such as the cardinalities of the natural numbers and rational numbers being the same, or of the natural numbers having a smaller cardinality than all real numbers in the interval of 0 and 1 that has the same cardinality as the real numbers (I have been able to understand the theory behind those but even so it still feels very counter intuitive to me).

So in trying to get a better intuition for it I though about a lot of examples that would derive from the theorem and it’s implications and ended up thinking of a counter example to one of the implications from Cantor’s theorem that I wasn’t able to figure out why it would be wrong.

The counter example I had thought of would be a bijective function that would link all natural numbers plus some rational numbers to all real numbers in the interval between 0 and 1.

The general logic of the function would be the following:

The input of the function would be a set spanning all natural numbers as well as a “few” rational numbers (I will better specify which are those rational numbers later)

So for this function, let’s call it F, we would have a certain value F(x) if x € N.

This value would be calculated by assigning each natural number (ex: 1,2,3,4,5) to a respective decimal between 0 and 1 (ex: 0.1, 0.2, 0.3, 0.4, 0.5) following the intuitive rule that you might have noticed between both examples.

If x is a natural number divisible by 10 though, such as 10 or 20, we would have a problem using the simple rule from above as 10 would lead to 0.10 and 20 would lead to 0.20, both of which can be simplified to 0.1 and 0.2. Both of these values had already been obtained from x=1 and x=2, which would mean the function is not injective. We can dodge this problem though by instead assigning the value of 10 to what would be the next decimal value(ex: 0.11), so we would be able to continue with 11>0.12, 12>0.13, 13>0.14 and so on( the “>” sign was used to mean leads to rather than it’s bigger than in this occasion).

Now following this rule we should have a function that leads all natural numbers to all decimal numbers between 0 and 1 that have a similar structure to how natural numbers are written.

There is still a problem t

I’ve been excited for this since I learned about it months ago, but I don’t think too many other people are aware... Either that, or they really REALLY dislike Rob songs.

https://www.google.com/amp/s/screenrant.com/ghost-and-molly-mcgee-first-look-rob-cantor/amp/

This Youtube video says the original version of Cantor's book "Contributions to the Founding of the Theory of Transfinite Numbers" was not published because publishers (and especially French and English translators) claimed it was too metaphysical and not grounded in enough mathematical rigor.

Is this true and is it possible for a scholar to go to Germany and discover and publish the original text?

Update: You can't address something that you aren't aware of, but you also can't correct definitions that are already consistent... So it looks like Cantor knew that there could be problems with allowing any collection of objects to be a set, but he certainly was not aware of what those problems could be. Frege, on the other hand (from what I read), stood behind his Basic Law V (naive comprehension), which states (in my own words - correct me if I'm wrong) that, given a property, you can create a set of all sets that each satisfy that property. This means that Frege was incorrect by believing Basic Law V, and Cantor was not inconsistent by being conservative in his definitions and by requiring that collections be well-defined from the get-go. I have no evidence of Cantor believing Basic Law V (I didn't look that hard into this though), and because of that Cantor was never wrong about his set theory - he just wasn't aware of how to formalize it. Neither of them knew what problems existed with Basic Law V nor were they aware of how to address things that had not existed yet. I have more questions that I will be asking on this sub. Thanks everyone for helping me understand this history.

Original question:

I translated one of Cantor's articles, and saw that he defined a set to be ~ a collection of well-defined objects ~ which does not seem to logically imply Russell's paradox, considering the notion well-defined can require ZFC-axioms, and so on, right? Cantor doesn't go on to explain what well-defined really means (at least as far as I read), but that's not really the point. There's a problem when you say that sets are collections of objects, since there are collections of objects that should definitely not be sets. I don't see a problem when you add the requirement that the collection is well-defined, since this removes ambiguity in whatever way you want, depending on how you want to remove ambiguity.

I notice textbooks use universes to try to solve the problem of naive set theory in an attempt to kick the philosophical can down the road into courses that are more suitable for discussion on ZFC-axioms, and Cantor's use of well-defined seems very analogous to that solution. Am I missing something? Was Cantor really the author of naive set theory? Was Russell solving a paradox that was already kind-of solved by Cantor here? Did Cantor know why he needed to use that wording well-defined? This makes me feel like naive set theory was never

I worked on this project for a few weeks, and tried synthesizing lots of concepts in set theory. Any feedback on how these videos can be improved would be great. All content is released under Creative Commons "No Rights Reserved", so please feel free to use this video however you want. Thanks to Nathanfenner for taking the time to comment on my last video. I hope this video is more productive for everyone.

https://youtu.be/UjklRuyFaIA

A friend of mine shared this problem with me awhile back.

Given a fat Cantor set C (say the middle 1/4ths Cantor set) considered as a subset of [0, 1], consider the distance function f(x) := dist(x, C) as a function on [0, 1]. This function is Lipschitz continuous and hence differentiable a.e. by Rademacher’s theorem. The problem is to determine all points in C at which f is differentiable.

I think I managed to find an explicit description of those points. I’ve written the result in more detail here, but if any of you would like to try it, then don’t open the link! I’m not a hundred percent sure the result is correct but it might spoil at least part of the solution.

So I have a homework problem basically asking for a bunch of proofs of some of the properties of the middle-fourths "fat" Cantor set, which ok cool that wasn't even mentioned in lecture. I turned to Wikipedia, which had almost nothing on it and thus I ended up on the page for the middle-thirds Cantor set. Of course since I don't have much background in this stuff and my class is just introductory, just one article isn't sufficient to fully understand the differences between the two, and I eventually got stuck on the page for a dense set, which defines it like this:

>In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A constitutes the whole set X.[1] Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A

However, that also happens to be one of the things my prof did define: like this. (Here's a link I rendered in latex that's more legible)

Both of these definitions make sense to me, but it seems like they're defining two completely different things. When I think about the physical concept of density, my prof's definition is very intuitive: there's essentially no "gaps" between the elements in the "dense" area(s) of the set. And I just don't see how this Wiki definition has anything to do with the closeness of the actual members of the set; it seems like it's more describing how thoroughly the subset is "touching" (for lack of a better word lol) the rest of elements of the topological space.

Normally I would just go with the prof's definition but it seems like it completely contradicts Wikipedia, which claims that the Cantor Set is both perfect (no isolated points) AND nowhere dense, two things that seem mutually exclusive to me. One of the things it says in their article on a nowhere dense set (also somethi

Can someone explain this to me? Intuitively it makes sense to me to say that its length approaches zero, but never actually becomes zero. It feels weird, considering it is uncountable. Any help would be greatly appreciated.

Edit : there was a simple misunderstanding, problem solved everyone :)

Hi people,

You know some way for created a Cantor set in Lisp?

I found a proof online, but I am not sure what kind of theorem is used here... Why it suffices to show for any open ball in [0,1], it can't exist in the generalized cantor set

https://math.stackexchange.com/questions/1492026/how-to-prove-complement-of-generalized-cantor-set-is-dense-in-0-1

For infinite countable sets, we can just use a table (assuming an enumeration exists) and the diagonal argument. But for non-countables, we have to resort to the written argument below:

https://en.wikipedia.org/wiki/Cantor's_theorem#Proof

But, if instead of P(A) we change it to P(A)-∅, that proof no longer works, right?

Because if we make B the empty set (by having a ∈ f(a) for all a ∈ A), then we don't need an a pointing to B. Since B would be ∅, and we're mapping to P(A)-∅

So I was wondering if there's a "better" proof, because just taking one element of an infinite set doesn't change its cardinality. And if we can't prove the impossibility of a surjection to P(A)-∅, then how is the proof for P(A) itself solid?

EDIT: On the other hand, conversely (and going in the opposite direction of the above), if by adding a finite number of elements (in this case just ∅), we can prove there is no surjection, then we just proved the greater cardinality of the target set... right?

So I'm going about it backwards... if we can prove no surjection possible if only a finite number of elements added to a set, then the greater cardinality is established... right?

Anyway, my understanding is so fuzzy and hazy, please chip in with your thoughts and insights

So one of my profs gave us a few voluntary exercises over the semester break and one of exercises are various proofs about the cantor set. I've managed to proof all of them but I'm a bit unsure about my proof of uncountability. I've done it via Cantor's second diagonal argument as follows:

>let C be the cantor set and (a_n)_(n ∈ ℕ) an C-valued sequence. We have:
>
>a_1 = (0,x_11 x_12 x_13...)_3
>
>a_2 = (0,x_21 x_22 x_23...)_3
>
>a_3 = (0,x_31 x_32 x_33...)_3
>
>...
>
>. We define d := (0, d_1 d_2 d_3 ...)_3 where ∀n∈ℕ:d_i = 2 if x_ii = 0, 0 if x_ii = 2.
>
>It follows that d∉(a_n) since it differs by each element of the sequence in at least one place, but d∈C since it's 3-adic expansion contains only zeroes and twos. It thus holds that there is no surjection from ℕ to C ⇒ C is uncountable.

So my problem now is that C contains all infinite 3-adic expansions consisting of 2s and 0s and thus also contains stuff like (0.20202222222...)_3 = (0.2021)_3. If I now take the C-valued sequence it's possible that one element is e.g. (0.2021)_3 and I make it different from that by e.g. setting the 1 to a 0 but maybe the series also contains the periodic expansion and I thereby construct exactly that representation of the same number.

Now you may notice that in my proof I defined d_i only for x_ii's of 0 and 2. My way of thinking was that I just take the digits (the x's) of each a to be the expansion ending in ...222... (if it is one of those special cases that is) - this can be done quite easily via a transformation I've done in another proof. I'm quite certain that this solves the problem but not 100% so I thought it'd be better to ask here :)

PS: I know that this problem can be avoided in the diagonal-argument-based proof of the uncountability of ℝ by selecting the replacement digits accordingly but I don't think this is possible here since we only have two digits to select from. And lastly: another idea I had was to define d_ii to be 2 if a_ii is 1 which would avoid constructing the periodic expansion of the same number but this has the potential problem of constructing the periodic expansion of another number in the sequence

[0,1]\C is the same as [0,1]⋂(union of Cn^(c)), so you just take the middle thirds removed when constructing the Cantor Set.

The boundary of those intervals (for ex {0, 1, 1/3, 2/3} are the points that remain in C, since they sit at the edge of their respective intervals and will never get removed.

So, is it correct to say that that Boundary in the title is equal to C for this reason?

Hi there, I am trying to work through some examples of cantor bendixson ranks of subsets of R, but I'm not really sure if I'm formalizing my ideas correctly. I'm also not sure I understand what it means for a closed subset of R to have cantor bendixson rank omega.

So, first I want to construct a set such that for a fixed r in R, the CB Rank is r+1.

First my intuitive understanding: I'm thinking inductively here, so basically at step 0 we want to have some point in our set so that CB rank is 1, so let our set be {0}. Now, I know how to go to CB rank 2, I can take {0} U {1/n : n in N}. This works because 0 is now no longer isolated, but each of these 1/n is isolated. So now inductively to go from step a to step a+1 we want to basically take any isolated point in step a and add a sequence of rationals which converges to that point in step a+1.

Now a bit more formally: In step r = 0, let X_0 = {0}. In step r + 1, let X_(r+1) = X_r + {A_i | i in N} where A_i is defined as follows: fix a_i, a_(i+1) in X_r such that a_i < a_(i+1). Now fix a sequence of rationals a_i < b_(i, 0) < b_(i, 1) < ... < b_(i, j) < ... < b_(i+1) such that b_(i, j) --> a_(i+1) as j --> infinity and A_i = {b_(i, j) | j in N}.

Effectively what I'm trying to do there is formalize the idea of fixing that sequence of rationals which converges to each isolated point, but I know I fear that in my attempt to be more formal I've just become less clear. If it's unclear or doesn't feel rigorous the way that I'm phrasing it, please let me know.

I'm also having trouble of conceiving of how to extend this process beyond the finite, because I don't really understand why you would need to take infinitely many CB derivatives. Would the omega rank set just be the union over n in N of each of these X_n? If this is way off please explain why, but this is the closest thing to what I can imagine, because I suppose you are essentially just taking this process and doing it infinitely many times, thus you must derive the set infinitely many times to get back to X_0?

I've tried to read more about this online, but everything related seems to be primarily through the lens of general topology, but I haven't taken a topology course so I don't really understand the language of homeomorphisms and embeddings into Q and other approaches I've seen for tackling this problem.

Finally, this is for homework so please don't just give solutions or answers, my primary goal is

I got 1/4 for the last question on it, but the second last was just unanswerable for me. If not for that and the induction proof I would've been golden.