What is Cantor: Definition and 72 Discussions

A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Christian worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds.
In Judaism, a cantor is one who sings and leads people in prayer in a Jewish religious service and may be called hazzan. A cantor in Reform and Conservative Judaism is an ordained clergyperson, similar to that of an ordained rabbi, if the cantor has gone through seminary training or been certified as a "Cantor" from an endorsed seminary.
"Cantor" is used as a translation of equivalent terms in other languages, such as for the leader of singing on a traditional Kerala snake boat, a Chundan Vallam. A similar term is precentor, defined as a leader of the singing of a choir or congregation.
Cantor is a Coptic word (language that very similar to Greek) which if translated to English means teacher.
More specific types of cantor include:

Cantor in Christianity, an ecclesiastical officer leading liturgical music in several branches of the Christian church
Protopsaltis, leader master cantor of the right choir (Orthodox Church)
Lampadarios, leader of the left choir (Orthodox Church)
Domestikos, leader assistant to the Protopsaltis of the right choir and/or to the Lampdarios of the left choir (Orthodox Church)
Hazzan in Judaism, a singer and/or musician. Orthodox Judaism only allows men to be cantors, while the other branches allow women. Reform Judaism and Orthodox Judaism ordain cantors from seminaries. Ordained cantors serve as clergy in their congregations and perform all ministerial rites as rabbis.
An ordained Muezzin, who calls the Adhan in Islam for prayer, that serves as clergy in their congregations and perform all ministerial rites as imams.

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  1. Kekkuli

    I Criticism of the doctrine of infinity

    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...
  2. C

    A Understanding Cantor set C in Ternary form with 1/n factor in front C

    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...
  3. A

    MHB Cantor Diagonalization | Find a Number Not on the List

    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…
  4. Cantor080

    I Was there any need or utility or aim, for Cantor's theory?

    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...
  5. F

    I Infinity: The Limit Concept and Cantor Transfinites

    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...
  6. Wendel

    I Bernstein-Schröder Theorem

    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...
  7. R

    I Understanding the Paradox of the Cantor Set: A Closer Look at Its Derivation

    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.
  8. EnumaElish

    I How is Cantor set similar to coin tosses?

    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?
  9. Jimster41

    B Calculus of Cantor Spaces (and the like)?

    Do Cantor (and similar) spaces (sets) defy traditional differential calculus? If so are there alternatives with similar capabilities?
  10. AlexDB9

    B Speed of Light, Rapidity & Cantor's Infinities

    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...
  11. slatts

    Is the Multiverse of Cantor Dust a Viable Solution to Olbers' Paradox?

    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...
  12. A

    Cantor set ℵ , inductive proofs by openly counting.

    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...
  13. H

    Hausdorff dimension of the cantor set

    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
  14. A

    Where Can I Find Cantor and Dedekind's Letters on Set Theory Development?

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

    Topological property of the Cantor set

    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
  16. A

    Cantor proof / infinite binary sequences

    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...
  17. P

    Is My Proof of the Cantor Bernstein Theorem Correct?

    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...
  18. T

    Cantor Set - Perfect and Totally Disconnected

    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...
  19. B

    Cantor set end points are and are not countable

    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...
  20. D

    MHB Cantor Set Numbers: Explaining the Unique Ternary Expansion

    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.
  21. D

    MHB Cantor ternary set, construction

    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.
  22. B

    Exploring the Cantor Set: Why There Are No Interior Points

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

    Cantor, infinity and cosmology

    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...
  24. C

    A way to count the uncountable Cantor Set.

    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...
  25. R

    Proving rationals cannot have dense orbit in Cantor set

    Homework Statement Let C be the standard Cantor "middle third" set (ie Ck = {x:0\leqT^{k}_{3/2}(x)\leq1} and C = \bigcap^{inf}_{k=0}Ck where T^{k}_{3/2} = 3x if x<1/2, = 3 - 3x if x \geq 1/2) Show that a rational number x = p/q \in C cannot have dense...
  26. J

    Baby Rudin - Cantor Set. A question.

    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?
  27. C

    One more question about the cantor set.

    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...
  28. C

    Understanding Cantor Set: Uncountability & Measure

    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.
  29. S

    Determining if Numbers are in the Cantor Set

    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...
  30. S

    Are 5/27 and 8/9 in the Cantor Set?

    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...
  31. J

    Newton's law with Cantor potential

    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...
  32. L

    Is the cantor set diffeomorphic to the fat cantor set?

    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...
  33. L

    Cantor sets, Fat cantor sets and homeo and diffeo

    Homework Statement Ok so the first part of the problem was constructing fat cantor sets, cantor sets with some positive lesbegue measure. The second part involved proving that any fat cantor set is homeomorphic to the regular cantor set. The third part asks whether there is a diffeomorphism...
  34. S

    How can I prove the properties of points in a Cantor set?

    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...
  35. B

    Cantor Set/Space and Retracts.

    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...
  36. B

    Closed sets in Cantor Space that are not Clopen

    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.
  37. K

    Proof of Cantor Set: Consider g'(x), f_n(c), and f'(x)

    Consider g(x)=x^2sin(1/x) if x>0 and 0 if x<=0 1. a) Find g'(0) b) Compute g'(x) for x not 0 c)Explain why, for every delta>0, g'(x) attains every value between 1 and -1 as ranges over the set (-delta,delta). Conclude that g'(x) is not continuous at x=0. Next, we want to explore g with...
  38. T

    Cantor Set: 7/12, 1/3, 1/4, 11/12

    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...
  39. T

    Homeomorphism of the cantor set to itself

    Homework Statement I feel like I got away easy with this one. Could somone let me know if I got it wrong? Thanks Is there a homemorphism from the Cantor set C to itself sucht hat for some x,y\in C f(x)=y Solution. Yes We know that the canot set is homemorphic to the space \left\{...
  40. R

    Integral of cantor function

    Homework Statement consider the ternary cantor set C, and the asscoiated cantor function f, and the associated Lebesgue-Stieltjes measure u. what is the integral of f over all of R with respect to u? Homework Equations The Attempt at a Solution i know that under the...
  41. nomadreid

    Infinitesimals in the Cantor set

    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)...
  42. B

    What is the topological characterization of the Cantor set?

    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!
  43. A

    Hausdorff dimension of the Cantor Set

    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?
  44. K

    Prove Sums of Cantor Sets in [0,2]

    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...
  45. Demon117

    Proving Fat Cantor Function is Non-Riemann Integrable

    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...
  46. M

    Is 1/4 in the Cantor Set? An Exploration of Proofs and Characteristics

    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.
  47. L

    Is the Cantor Set the Only Example of a Totally Disconnected Set?

    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...
  48. S

    Measure theory and Cantor function

    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...
  49. K

    Generalized Cantor Set

    "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...
  50. daniel_i_l

    Cantor normal form multiplication

    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...