Cantor Definition and 68 Threads

Hi, PF Theorem The set ##\mathbb{Q}## of the rational numbers is countably infinite. Proof The rational numbers are arranged thus...

On the one hand, Cantor showed that not all real numbers can be enumerated, while on the other hand he showed that rational numbers can. Cantor demonstrated this with a grid. In the picture below, a natural number (yellow) is assigned to each rational number in order, but since the natural...

Dear Everybody, I am confused by ##1/n C##, where C is a cantor set in base 3 and ##n\geq2##. I can understand the construction of the normal Cantor set. How do I comprehend this set with this extra condition. Do I multiply the set with ##1/n## or not? Thanks, Cbarker1 mentor note...

Consider the following list of numbers. Using Cantor's diagonalization argument, find a number not on the list (use 2 and 4 when applying Cantor' argument). Give a brief explanation of the process. 0.123456876… 0.254896487… 0.143256876… 0.758468126… 0.534157162…

Was there any need or utility or aim, for which Cantor created his theory? Did Cantor's theory clear any of the problems which existed before? (Though my user name is Cantor, I don't know lot about him or his theory :biggrin:) Reddit...

Supposedly, infininity has been purged from mathematics. Both the infinitely small and the infinitely large have been replaced by the idea of a "limit." For example, a series x0+x1+x3+... is not considered to be a literal infinite sum with infinite terms but only the limiting value of an...

The theorem: Let ##X##, ##Y## be sets. If there exist injections ##X \to Y## and ##Y \to X##, then ##X## and ##Y## are equivalent sets. Proof: Let ##f : X \rightarrow Y## and ##g : Y \rightarrow X## be injections. Each point ##x \in g(Y)⊆X## has a unique preimage ##y\in Y## under g; no ##x \in...

I am puzzled by the derivation of the Cantor set. If the iteration of removing the middle-thirds leaves an uncountable set of points, it seems the iteration had to be performed an uncountably infinite number of times. Is this the case? If so, that seems paradoxical to me.

The wikipedia page on the Cantor set states that it is a model for an infinite series of coin tosses. In what sense are recorded coin outcomes similar to a set of points on the real line?

Do Cantor (and similar) spaces (sets) defy traditional differential calculus? If so are there alternatives with similar capabilities?

The natural expression of speed in relativity (and thus the true meaning of speed) is through the concept of rapidity, which comes from incorporating the gamma factor. It turns out that the rapidity of light is infinite. So the question of whether there can be speeds greater than light becomes...

At my Wikipedia-minus-math level of understanding, the problem with any resolution of "Olbers' Paradox" through a fractal distribution (such as the "Cantor set" depicted in the Wikipedia article of that name) of stars or star clusters, rather than the alternative of a beginning of our multiverse...

I have been looking at the idea of 1:1 correspondence as a method of determining set size/cardinality, and have noticed that the principle allows for inductive proofs, which I think are properly constructed, that can come to conclusions which are clearly wrong under traditional set theory if...

Hi, Using the definition of Hausdorff measure: http://en.wikipedia.org/wiki/Hausdorff_measure I am looking for a simple proof that Hd(C) is greater than 0, where C is the Cantor set and d=log(2)/log(3) Thank's in advance

Does anyone know where to get the letters(English translation) between Cantor and Dedekind during the development of set theory??

Let X be a metric separable metric and zero dimensional space.Then X is homeomorphic to a subset of Cantor set. How can it be proved? Thank's a lot, Hedi

Hi, I've been reading a textbook on set theory and came across Cantor's proof of the statement that the set of the infinite binary sequences is uncountable. However there is one thing that is not clear to me: The nth such sequence would be: An = (an,0,an,1,...), n = 0, 1, 2,... where...

Hi guys, I've got some problems with the cantor bernstein theorem. I'm having a hard time with all the proofs I've found, but I've actually come up with a proof myself... it will be no doubt wrong in some part though, so it would be great if you could check it for me and tell me what's wrong...

I am getting ready for grad school in the fall, and re-teaching myself a bunch of undergraduate subjects. Right now I am reading up on the topology of the real number line. I have come across a fact that is really difficult for me to wrap my head around: The Cantor set is both perfect, and...

The number of end points of the cantor set double each time an iteration is performed, therefore the total number of end points after infinite iterations is ~ 2^N where N is cantor's aleph null. 2^N is, however, c (the number of the continuum) and is therefore uncountable but we know that the...

Explain why the Cantor set consists precisely of all the numbers between 0 and 1 (including 0 and 1) which can be represented by a ternary expansion in which the digit 1 does not appear anywhere in the expansion. I believe this has to do with always taking a 3rd away.

Explain why all the end points of the closed intervals that comprise $C_n$ are in the Cantor set $C_n$. I understand why this is true but I don't know how to explain it.

Why there is no interior points in a Cantor set? Please explain me in detail.

I should start a new thread for my questions rather than hijack others... Posted this on another thread but it didn't get any response, so bear with me if you've seen it. Has anybody looked into Cantor's works on infinity and seen how they relate to the question of an infinite universe...

It's not that I discovered a way to count it or anything, but I think I have some confusion about it. I understand that Cantor set isn't countable and I accept the proof also. But, what if we count the elements of the set like the following? 1, 0, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, 1/27...

I do not get the second sentence of the paragraph in the image. What segment does he refer to when he says "no segment"? And why is it 3^-m < (beta - alpha)/6? Why 6?

Lets start with a line segment from zero to 1 and instead of removing like the middle 1/3 can we remove an infinitesimal amount, and then keep doing this forever. It seems like this set would still have measure 1. Unless I don't understand measure or infinitesimals. And if we looked at the line...

How can the cantor set be uncountable and have zero measure. Couldn't I map the cantor set to another uncountable set that did not have zero measure. I probably don't understand measure or the cantor set very well. Any input will be much appreciated.

Homework Statement I have two numbers: 509/729 and 511/729. I want to determine if they are in the Cantor set. The Attempt at a Solution I have: 509/729 in base 3 is: 0.200212 So this is not part of the cantor set because it can't be expanded in base 3 using only 0 and 2...

Homework Statement I am trying to find if 5/27 and 8/9 are in the Cantor set. Homework Equations C_2=[0,1/9]\cup[2/9,3/9]\cup...\cup[8/9,1] C_3=[0,1/27]\cup[2/27,3/27]\cup[4/27,5/27]\cup...\cup[26/27,1] The Attempt at a Solution I have: 8/9=(0.22)_3 and it is an endpoint in one of the...

I want that [0,\infty[\to\mathbb{R}, t\mapsto x(t) satisfies \ddot{x}(t) = -\partial_x U(x) where U:\mathbb{R}\to\mathbb{R} is some potential function. Then I set the initial conditions x(0) < 0, \dot{x}(0)>0, and define U(x) = \left\{\begin{array}{ll} 0,&\quad x < 0\\ \textrm{Cantor...

Ok I originally posted this question in the homework section but it doesn't seem like anyone knows the answer there. Hopefully someone can help me out here! 1. Homework Statement Ok so the first part of the problem was constructing fat cantor sets, cantor sets with some positive lesbegue...

Homework Statement Let C be a Cantor set and let x in C be given prove that a) Every neighborhood of x contains points in C, different from x. b) Every neighborhood of x contains points not in C Homework Equations How can I start to prove? The Attempt...

Hi, All: I was thinking of the result that every compact metric space is the continuous image of the Cantor set/space C. This result is built on some results like the fact that 2nd countable metric spaces can be embedded in I^n (I is --I am?-- the unit interval), the fact that there is...

Hi, Is there a characterization of subsets of the Cantor space C that are closed but not open? As a totally-disconnected set/space, C has a basis of clopen sets; but I'm just curious of what the closed non-open sets are.

Homework Statement Which of the following are in the Cantor set: 7/12, 1/3, 1/4, 11/12? Give the ternary expansion of each. The Attempt at a Solution I see that 1/3 is in the Cantor set and has a ternary expansion: 1/3 = 0/3 + 2/3^2 + 2/3^3 + 2/3^4 + ... I am fairly certain that...

I am not sure into which rubric to put this, but since there is some Model Theory here, I am putting it in this one. First, I define the Cantor set informally: A(0) = [0,1] A(n+1) = the set of closed intervals obtained by taking out the open middle third of each interval contained in A(n)...

Hello PF! Was wondering if anyone knew a good reference on the topological characterization of the cantor set, proving that if a metric space is perfect, compact, totally disconnected it is homeomorphic to the cantor set. Thanks!

Hi everyone! I am thinking about, how can calculate the Hausdorff dimension of the Cantor set? I know, that this dimension is \frac{\log 2}{\log 3} but I cannot prove it. Any ideas?

I'm supposed to show that the sum C+C ={x+y,x,y in C}=[0,2] a) Show there exist x1,y1 in C1 for which x1+y1=s. Show in general for any arbitrary n in the naturals, we can always find xn, yn in Cn for which xn+yn=s. b) Keeping in mind that the sequences xn and yn do not necessarily converge...

I have been thinking about this for quite some time now. When I look at the function that descibes the fat cantor set namely: f(x) = 1 for x\inF and f(x) = 0 otherwise, where F is the fat cantor set. I wonder, how do I prove that this is non-riemann integrable? I have considered...

Can anyone show me a proof of 1/4 being in the cantor set?? My prof said it is, I read it is, i saw no proof though. Not in my text anyway, also couldn't find it on google.

I'm having trouble understanding the Cantor set. The idea of a nowhere dense uncountable set makes no sense to me, because of the following argument I thought of: Let x be any element of the Cantor set. Since the Cantor set is nowhere dense, there exists some open interval I_x containing x...

Homework Statement Show that there is a continuous , strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero Homework Equations The Attempt at a Solution I need to find a mapping to a countable set or cantor set but I...

"Given (rn), rn E (0,1), define a generalized Cantor set E by removing the middle r1 fraction of an interval, then remove the middle r2 fraction of the remaining 2 intervals, etc. Start with [0,1]. Take rn=1/5n. Then the material removed at the n-th stage has length < 1/5n, so the total...

Let's say that a is an ordinal and it's cantor normal form is: a = {\omega^{\beta_1}}c_1 + {\omega^{\beta_2}}c_2 + ... I read that a \omega = {\omega^{\beta_1+1}} But I couldn't find a proof anywhere. Can someone give me a source or point me in the right direction so that I can prove...

if you go down to the section called "do they always exist?" you'll find a venn diagram for 6 sets: http://www.combinatorics.org/Surveys/ds5/VennWhatEJC.html would a horizontal cross-section resemble a cantor set, if there were infinitely many sets? it looks like it would vaguely resemble...

sorry for starting yet another one of these threads :p As far as I know, cantor's diagonal argument merely says- if you have a list of n real numbers, then you can always find a real number not belonging to the list. But this just means that you can't set up a 1-1 between the reals, and...

What is the definition of a fat cantor set? How do I show that the fat cantor set has positive Lebesgue measure and does not contain any interval. I know for the cantor set that at each stage, we remove the middle third of each interval starting with [0,1]. I am wondering if instead for the...

This is a statement my professor made in class some time ago (as a means to show that C contains a Hamel basis) that seemed fairly innocent, but it's bothered me for awhile. I did some searching online, and it seems that C+C=[0,2]. There it was again stated that this is fairly easy to show...