Talk:Cantor set - Knowledge

464:

did not feature the numeral one. So, base three decimal expansions of for example 0.02202 and 0.202 would be included, while 0.22221 and 0.11201 would be excluded from his set. Taking all these base three numbers, Cantor ingeniuosly directly compared the numbers in his set 'one ot one' to EVERY number on the number line in base 2. So, from above 0.02202 in base 3 becomes in base 2 0.01101, and 0.202 becomes 0.101. So there is a 'one to one' partner for every single decimal expansion in base two from his base three set, even though he removed nearly all the numbers. So he demonstrated a part can be equal to the whole, and demonstrated some unrecognized at that time property of infinity.

3355:

which contain the digit 1 at the first position. Let's now allow this interval, as long as the number does not contain any more digits 1; this replaces the omitted interval with a scaled down Cantor set. Now we have three intervals (for numbers with digit 0, 1, or 2 at position 1), each omitting the central subinterval (digit 1 at position 2), which we replace with a scaled down copy of the Cantor set (no more digits 1). Now we have nine intervals (all possible digits at positions 1 and 2), each with an omitted central subinterval which we replace with a scaled sown copy of the Cantor set, etc.

352:?? The reference to Cantor's paper is given in "historical references" at the bottom of the article. You can get the book on amazon for about $ 8 or so, and most university libraries will have it. It is a rather mind-opening read, espcially if all that you know about the Cantor set comes from the crappy descriptions given in pop-lit books on fractals (which is all I knew when I embarked on this journey). I'm guessing most books on topology will also discuss it; I don't think they do the "middle third" contruction either. 851:.) This example of 1/4 was mentioned in an earlier version of this article, but it was muddled and misstated (at least one version said "1/3" instead of "1/4"). Similarly, the Cantor set's inclusion of 7/10 is mentioned briefly, later in the article. Since it's somewhat surprising that the Cantor set includes points besides those endpoints (and a common misconception that it doesn't), I think it's important that we mention it more explicitly and explain how these "extra" points made it into the Cantor set. -- 95: 85: 64: 31: 190: 169: 2015:

but only imagenary. Ok but then we don't know what will be result of that imagenary procedure. In particular we cant know that it will be the number 1. Now it may seem intuitive, but i still say this is mathematics and a proposition is considered to be true only after a rigorous proof. And i dont see how we can apply any formal thinking about an informally described imaginary procedure, let alone to prove that it will result in something partiular.

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interval (0,1) which do not have a base-3 representation containing the digit 1". (I am not sure why does the original definition use the convention that the endpoints of intervals are included rather than excluded; in any case, all endpoints are rational numbers, so by excluding them we get a set that differs from the original by countably many members; so the restricted set is still uncountable and has measure zero.)

1245:(for the first digit being 0, 2, or 3). Now let's remove numbers which contain the digit 1 at the second position. Again, this removes one quarter of each subinterval, and so on. When we look at the progression of measures of the partially constructed sets, we get 3/4, (3/4)^2, (3/4)^3 and so on; the limit of this sequence is 0. Finally, the finished set is a subset of each partial set, so it has measure 0. - 1989:

with the digits as is done in the presented proof". What is arbitrary about it? Why are we not allowed to do it? Then we are told that "0.999999.." "is NOT a valid (rigorous) definition of a number". Why not? It means the number which is the limit of the geometric series 0.9 + 0.09 + 0.009 + ..., as the same post had previously stated: there is nothing invalid about that.

22: 2044:; that is, the number 0.999..., i.e. the number with a nine at every decimal position after the decimal point, is by definition equal to the infinite sum 0.9+0.09+0.009+... And the infinite sum is the limit of the sequence of partial sums: 0.9, 0.99, 0.999...; and the limit happens to be equal to 1. (For the same reason, the number 0.111... is equal to 1/9.) See 4329:

Smith–Volterra–Cantor set where the middle (b-2)/b is removed can be represented as all real numbers from 0 to 1 that can be written in base b with only 0s and symbols representing b-1. A corollary would be that the Cantor Set can be defined as all real numbers from 0 to 1 that can be written in base 3 with only 0s and 2s. Here is a non-rigorous proof of this:

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However, I don't see why it is necessary to bring up the Cantor-Bernstein-Schroeder theorem. If the Cantor Set's cardinality is greater than or equal to the cardinality of and also less than or equal to it, then it follows that they are equal. If the theorem isn't directly related here, then maybe

1948:

Now the problem: what does 0.9999999... mean? In mathematics we can not work with an object unless we define it first. If we define 0.999999.. to be the limit of 0.9+0.09+0.009+... (which is equal to 1) then 0.999999.. is 1 by definition, but then we are not allowed to do the (arbitrary) manipulation

1066:

In general, for a base-n set, there are a finite number of sets possible to construct this way. For decimal, I believe there is 9+9*8+8*7*6+7*6*5*4+6*5*4*3*2 possible solutions. It might be something else but I'm too tired to double check this. Please verify. If instead of a recursion, each cycle

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0 1/3 2/3 1 ================================================================================= =========================== =========================== ========= =========

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more often to denote any topological space homeomorphic to the basic middle-thirds example than that example itself. So even though this article mistakenly pretends that the term properly refers to the middle thirds example and that everything else is some kind of special use of the term, things are

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You’re trying to say we don’t NEED the digit 1 to express the full set. But the way it's written is poorly conceived and ambiguous. Yes, the numbers CAN all be expressed with 0s and 2s, but they’re not ONLY expressible with 0s and 2s. An infinite amount can ALSO be expressed using the digit 1. Using

1956:

We can not just say that "0.999999.." is the real number that when written in decimal system has has a zero, a point and infinitely many nines. This is NOT a valid (rigorous) definition of a number. If we say it has (for example) one million nines, then it will be ok. But we don't know what it means

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I would love to see a section in this article which discusses applications if there are any. See the article on eigenvalues and eigenvectors for a really good example of what I'm talking about. I know this is an important topic without applications, I just think it makes math more accessible to us

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Julian F. Fleron from a Note in Mathematics Magazine, Vol. 67, No 2 (Apr 1994) writes that his first reference to the Cantor ternary set is in a footnote to the statement that “perfect sets need not be everywhere dense” on page 575 of his original paper . She quotes him describing it as: The set of

3354:

Recall that the Cantor set is the set of numbers from interval from 0 to 1 whose base-3 representation does not contain the digit 1. Let's extend it to a set of numbers which contain finitely many digits 1. Start with the Cantor set; note that the set omits the central interval representing numbers

2014:

b) the second way to define it is "the real number which when written in decimal system has a zero, a point and infinitely many nines." Now this is not valid because it is simply impossible to write infinitely many nines. Now you may say that this (infinite) writting is not actual physical writting

2010:

a) if we define it to be the limit of 0.9 + 0.09... then this is just 1. Then we effectively define 0.9999... := 1. As i said the digits then bear no meaning. We could as well define 0.777777... := 1. That does not give us the liberty to do with the digits 777.. anything. I hope this time i managed

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It doesn't seem the article hits on the significance of his discovery in being that the axiom "The sum is larger than any part" is incorrect, but as he demonstrates they can be shown to be equal. The underlying work, Cantor working in base three, used only those base three numbers for his set that

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Repeating this indefinitely results in a summation across each interval, and since each corresponds to a digit of the number in base b (either 0 or b-1) because each interval is 1/n of the previous one in the previous iteration, every number in this Smith–Volterra–Cantor set can be represented in

1988:

I have read the above post several times to try to work out what the problem is, so that I can help to clarify the issue. However, it is very unclear, and I am not really sure what the misunderstanding is. The crux of the matter appears to be "we are not allowed to do the (arbitrary) manipulation

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Well, let's try it with fraction 1/4. Let's define our set as "The set of numbers from interval whose base-4 representation does not contain the digit 1. So let's first remove numbers which contain the digit 1 at the first position. This removes 1/4 of the full interval and leaves 3 subintervals

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Looking at the sets generated by removing different digits or different groups of digits is a nice idea for some recreational research. I can think of several interesting questions that would be fun to think about. But I don't think it needs to be in the article, since these various constructions

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Mr. 4 numbers, I didn't understand your addition. Did he have in mind a CONCRETE example of a series diverging on the cantor set (as is implied by the word "particular")? I find it hard to believe, such series were found much later. Maybe you are confusing this with the problem of uniqueness (see

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Yes, most topology textbooks describe the middle thirds construction as the introduction to the Cantor set. But then they define "a Cantor set" as any topological space that's topologically equivalent to it. Then comes the beautiful characterization of it as any topological space that is compact,

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To see this, we show that there is a function f from the Cantor set C to the closed interval that is surjective (i.e. f maps from C onto ) so that the cardinality of C is no less than that of . Since C is a subset of , its cardinality is also no greater, so the two cardinalities must in fact be

1366:

We could easily define the set to exclude the endpoints of each interval (like 0, 1, 1/3, 2/3, ...) By doing this, we would get not "a set of numbers on closed interval which have a base-3 representation not containing the digit 1" (the original Cantor set), but rather "a set of numbers on open

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Where do you find anything like that in that section? I don't see anything that says the Cantor set is finite; I'd have corrected it if I had. And: "countably finite" is a redundancy. There is a difference between countably infinite and uncountably infinite, but all finite sets are countable.

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I deleted ths section, as the proof was inadequate. But I think the point is worth making, so I used it to introduce the paragraph showing that the Cantor set is non-empty, without giving a complete proof. (One could give a proof without any measure theory by showing that if I is an interval of

1280:

The equation for the series sum is given incorrectly as (1/3)*(1-2/3) = 1 ... (1/3)*(1-2/3)=1/9 which is not the sum of the gaps. The series sum equation should be (1/3)/(1-2/3) = 1. I have no idea how to show that via math markup, I noodled around but kept getting parsing errors. Thanks! -

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I think the whole topic can be incredibly confusing if the reader has no background in topology. One can, for example, have finer topologies on the real numbers, and these other topologies are "worse" and have weirder weirdneses. This is why the standard topology is popular. This article would

966:

It also becomes obvious, that some irrational numbers belong to the set. Those are of course all numbers given with infinite, non-periodic ternary representations, built solely with digits 0 and 2. It's not so obvious, however, how to express any of those numbers in other systems, eg. binary,

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Now it is obvious, that Cantor set in uncountable, because its subset is same cardinality as the interval. To proove it replace digit 2 with 1 in the first part of regular expression above, and you get all infinite binary strings, which gives an injection of the interval into the Cantor set.

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It's been a while since I have thought about this, but I am pretty sure if you take out less than 1/3 of each segment, you end up with a set of nonzero measure. It's a simple geometric series; I don't have time now to go through it but maybe I will add that some other time if no one else is

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According to the current version of the article, a Smith–Volterra–Cantor set where the middle 8/10 is removed at each iteration can be represented as all real numbers from 0 to 1 that can be written in base 10 with only 0s and 9s. I think it is possible to generalize this fact to that a

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It is not necessary to always take the middle interval, either. For example, take the set of numbers whose base-3 representation does not contain the digit 0. It can be seen that this set still has measure 0 (though the construction is not as symmetric as the original Cantor set). -

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The best way to prove this is to just show that the Cantor set is in one-to-one correspondence with a countable sequence of binary choices. (The Cantor set is in fact not only in bijective correspondence with such a sequence, but it is even topologically equivalent to the countable

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So altogether at each layer there are finitely many sets of measure zero, so each layer has itself measure zero. And a union of countably many layers of measure zero still has measure zero. We have now gotten an uncountable, dense subset of interval from 0 to 1 of measure zero.

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Is it possible to say that all irrational numbers in the Cantor set are transcendent? I have this intuitive idea that algebraic irrational numbers represented in any basis should look like a random sequence of digits, and Cantor set elements in base 3 would lack number 1.

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I've restored the comment that you erased, but I've weakened it a bit so that it says only as much as is known to everyone who's read a bit about the history of Cantor's theory. For all I know it may be right, but it's been a long time since I've read anything specific.

2011:

to explain what i had in mind. What other example to give. Lets say you define o := 1 (where "o" is the letter and not the digit zero) then we take the liberty to treat the letter "o" as the digit "0" just because there is a visual resemblance. And then we get to 1 = 0.

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I still doubt that he was interested in Fourier series. As a kind of compromise I changed it to "trignometric series" which is ambiguous, it can refer to either Fourier series or uniqueness problems. But it still links to Fourier series, I didn't change it to link to

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The number of solutions is factorial according to my calculations. Technically they are just numerical representations of Cantor Sets. The base-3 set is just the most famous. It would probably be redundant though, to add them, since the general idea is the same.

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The image in the page (Cantor_set_in_seven_iterations.png) prints solid black from my (Linux) system, although it displays properly. Has anyone else encountered this? Would there be an objection to the image being modified so that the transparent bits are white?

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Oh crap! That's what comes of typing tired... I meant INfinite and Countably INfinite. I do think Countably Infinite is justifiable and a useful thing to say. So I stand (gently) corrected - in my typo's. But should we edit to add the word "Countably"?

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This is my first section on a talk page, so I do not exactly know what goes in here, and I am judging from other sections. Also, I have only done 10 article edits. You are free to edit this section to more reflect the ideals followed on the talk page.

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Just logging what I have done here. There were a few problems with the page, which I started fixing and then got carried away. The overall structure is unchanged, but paras have been rewritten for clarity, completeness and/or correctness. Summary:

1044:

In the article: "The Cantor set is the prototype of a fractal." Am I right in thinking that most examples (Mandelbrot, etc.) of fractals are continous and that a Cantor set isn't? This would be a useful consideration to anyone studying fractals.

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The Cantor set is not "extremely abstract". A strongly inaccessible cardinal is abstract. A functor on sheaf categories is abstract. The Cantor set is a simple, well defined set, for which it is normally easy to check if a specific point is in or

319:. He constructs the thing in several ways. I don't think he even mentions the bit about "removing the middle third" except maybe half way through the paper, as an example, and then, only in passing. At least that's how I remember it going. 3658: 1963:

Now that i think of it i even doubt that there is a bijection between the cantor set and the interval (and so both cardinalities are equal). It both goes against my intuition and lacks a (valid) proof (at least i know of no such proof).

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Before, I claimed that the Cantor set is homeomorphic to the p-adic integers; now I'm not so sure and I played it safe and replaced "p-adic" by "2-adic". Does anybody know if the 3-adic's are homeomorphic to our Cantor set? --AxelBoldt

1957:

to write "infinitely many" nines. For this purpose they invented the infinitesimal calculus a few centuries ago so we can do these things right. Then again, as i said, if we take the limit of a sequence, it wont work in this case.

1952:

To clarify, this is because by this definition "0.99999..." is just a different way to write 1, as is the word "one" and the digits in "0.999999..." are NOT digits in the mathematical sense anymore, but just like letters in a word

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Examples in base-10 (decimal), disallowing 1: 0.023495967029=in the set 0.2121=not in the set 1/9th=0.11111...=not in the set 1/8th=0.125=in the set Note that 'the set' is to be taken in context as being the set I just defined.

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I don't mind that the majority of the article is devoted to a simple example that is pounded to death; but the intro should at least give the correct definition, and note that the "middle third" construction is just an example.

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The Cantor Set is not only a mathematical abstraction to demonstrate "the infinite", but also a classical model and perhaps the simplest fractal with a complete and consensual documentation. As real-world Cantor Set (a linear

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I changed "named after" to "invented by", since that information is not otherwise given until the end of the article. "named after its inventor" might even be better, but with "German mathematician" it gets wordy. --anon

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This is irrelevant. The point is that the repeating part can only consist of 0s and 2s, but that the non repeating part can also consist of 1s. It has nothing to do with whether the period is equal to or greater than 1.

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is in the Cantor set, but clearly it's not an endpoint of any interval in any (finite) iteration. (1/4 is rational, of course, but it's easy to see how an irrational number could be similarly constructed: for example,

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In arithmetical terms, the set of 3 digit numbers that alternate between 1 and 3 (131, 313) consists of all integers that are expressible as a quaternary sum using only each digit to the power of itself.

2760: 4226:, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it is homeomorphic to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case 583:========= ========= === === === === === === === === = = = = = = = = = = = = = = = = 296:

A Cantor set is a compact perfect set with empty interior. A dynamically defined Cantor set is a Cantor set that may be defined by a family of contacting map (see for example the book by Palis-Takens

763:, is not a null set; you can build Cantor sets with any measure. For example, if you remove the rationals from the reals, you get a set that is is homeomorphic to the Cantor set but has measure one. 959:

First part describes all strings wih no 1's, the next one allows finite strings ending with 1 (after which only zeros are accepted), the last part describes the biggest, rightmost point of the set.

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These changes will obviously need some third-party input, or mass-reversion, or whatever. IMHO now, today, this minute, I have probably improved it, but I may think differently on re-reading it.

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zero set proof is pretty simple but some others would have to be shown as corollaries. i'm new at adding things to math articles that require computation symbols, how would go about adding them ?

4155: 300: 4051: 1793: 151: 868:: on the contrary, they were never removed; the construction did not remove them. Furthermore, there are uncountably many of these interior points, whereas the boundry points are countable. 2200: 3029: 4391: 3297:

that the intention is not the finite cyclic group Z/qZ, because in that case the countable direct sum of copies of this group is only countable, but its Pontrjagin dual is uncountable.

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that I am familiar with), then this statement seems wrong, also — and in any case, "q-adic integers" makes sense only if q is a prime number — which is clearly is not specified to be.

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is the least infinite ordinal, i.e., that of the natural numbers; and where the cardinal |C| is represented by the least ordinal having its cardinality.) So it is even possible that

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It's also kind of amazing that a set that's totally disconnected is also dense-in-itself (every point is a limit point), since the two properties seem almost contradictory.

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In the first step we delete the (1/3,2/3) interval, so we remove all numbers with digit 1 at the first fraction place, that is numbers 0.1... — except 0.1=1/3 itself. So C

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The introductory paragraph and the closing "historical remarks" have virtually identical sentences. Someone with time on their hands might want to tidy up the redundancy.

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Let us note that this description of the Cantor set does not characterize the complement of the Cantor set exactly, since the sets given by the formula are not disjoint.

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Cantor set is the set of numbers on whose base-3 expansion does not contain the digit 1. For example, it contains the following number in base-3: 0.202002000200002... -

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This is nonsense, since the formula given is an explicit formula for the complement of the Cantor set. Thus it characterizes the complement of the Cantor set exactly.

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obviously the cantor set will contain irrational numbers, but it seems only the end points are left. can some1 gimme an example? cuz i dunno how 2 prove there is one

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In arithmetical terms, the Cantor set consists of all real numbers of the unit interval that are expressible as a ternary fraction using only the digits 0 and 2.

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The section with all the properties of the Cantor set would be a lot more readable, if each property were listed before its proof rather than after its proof. --

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I am going to answer myself: the reason why the endpoints are included rather than excluded are probably its topological properties, so that the resulting set is

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In the second step we remove numbers, which have digit 1 at 2-nd fraction place, except 0.01 (1/9) and 0.21 (7/9), which are right ends of two sub-intervals. So C

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Uncountability: this section went straight into discussion of ternary characterisation of C - I added a preliminary statement as to why you would want to do this.

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does have a meaning: It is the cardinality of the integers (or equivalently, of the positive integers). It is known that the cardinality of the Cantor set C is

717:

Totally disconnected doesn't follow from nowhere dense in general. (TD is a property of a top space in itself, whereas nowhere dense requires a superset space.)

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Considering the left portion of one of the intervals , we can repeat this process in the second iteration, with that interval of length 1/n, or in base n, 0.1.

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non-zero length then it must contain an end point of one of the intervals removed in the construction. But this seemed a bit fiddly to put right at the start.)

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The Cantor set can be characterized by these properties: every nonempty totally-disconnected perfect compact metric space is homeomorphic to the Cantor set.

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This section mentions the Cantor Set is finite. I'd say it's COUNTABLY finite. Wondering whether to edit this section to add that detail. Might be TMI. :-)

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oh btw, how is it nowhere dense? it's dense in itself isnt it? i mean since its perf every point in it is a limit point, so cantor set S is dense in S...

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Actually, all the books of topology I know do just that. In fact, all sources I know use the term "Cantor set" to denote the "middle third" construction. —

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I sincerely suspect this proof to be fundamentally invalid and can't be made valid with minor corrections or additions though i can be wrong for this.

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And perhaps the most didactic and most used abstraction, it is not the Cantor set! The ref1 named it "hierarchical tree of the Cantor set". The tree

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finite or infinite representations consisting only of 0s and 2s belong to the Cantor set. These are more than those with repetends 0 or 2. E.g.

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digit, and it is their last non-zero digit. One might describe the possible ternary representations with a string regular expression like this:

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Characterisation of set in ternary: made links between ternary construction and existing material on numerals, including illustrative examples.

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Could you give a link to a reference, cause I can't find anything about his original definition online. Well.. if you ever revisit this site..

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whimsical notation to express the set with only 0s and 2s is a choice, not a limit of the set, and it’s certainly not an “arithmetical term.”

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Despite the massive number of adjectives, (nearly) every one of them had already been linked somewhere above. This is good cross-referencing.

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real numbers of the form $ $ x= \frac{c-1}{3} + \dots + \frac{c_v}{3^v} + \dots $ $ where $ c_v$ is $ 0$ or $ 2$ for each integer $ v$ .

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Hmm, maybe I'll look it up at the library here. But I kinda doubt I have enough background on the subject. Why don't you correct this page?

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probably benefit from a review of general topology, because merely applying "naive intuition" to the Cantor set can lead one into trouble.

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Using the idea of an arbitary base with one value disallowed for all digits, it is possible to just create an infinite number of sets.

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or something. I would still like if somebody could check this point in a history book. Or maybe just to unlink it until we are sure?

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If someone wants to describe the complement as the disjoint union of open intervals, that is fine. But the statement is nonsense.

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The article mentions six steps. It is much easier to illustrate six steps than it is to illustrate an infinite number of steps.

1795:. I would change it but I'm not sure what's the best way to explain the strong sense in which the sets are homeomorphic. — Carl 970: 707:

Mapping from C to : it wasn't explicit that this function is well-defined and onto. It now is. I also note that it is not 1-1.

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At layer 1 one scaled down copy of the Cantor set. (The set of numbers whose last appearance of the digit 1 is at position 1.)

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The relation currently given here is true for pretty much any set of points on the real line. What is needed here is that

1945:

Let's for simplicity work in decimal system; Similar of the above statement in decimal system would be: 1 = 0.99999999...

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is acceptable because the sequence of sets is monotone, so it is convergent in some sense: if one associates to each set

2007:

I duscussed about 2 possible ways to define the 0.99999... and explained what are the (different) problems in both cases:

1081:

have not yet proven to be of general interest in the areas of mathematics where Cantor sets are commonly studied. — Carl

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of the form k/3^n. (Unfortunately, I don't know of a way to construct one of these points.) I have removed this claim.

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is countable, the Cantor set is uncountable. So there are (uncountably many) numbers in the Cantor set that are not in

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At layer 2 three copies of the Cantor set. (The set of numbers whose last appearance of the digit 1 is at position 2.)

864:

I agree with your last statements, although I am not sure how to best explain it. Your words are slightly misleading:

3493:

As it stands, the sentence only serves to call attention to its own awkwardness and should be rewritten for clarity.

4332:

Lets suppose a Smith–Volterra–Cantor set where the middle (b-2)/b is removed and all numbers are written in base b.

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For example, a countable disjoint union of Cantor sets is also totally disconnected and perfect, but not compact.

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For now, yes. Maybe I'll add some things later if I look up specifics that I haven't read about for a long time.

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of the discrete space of two points.)But it is unknown, and some would say unknowable, where the cardinal number

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Hi, everybody, I'd like you to consider what exactly are ternary representations of the Cantor set points. Let C

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I feel a little out of my depth, but I believe the following equation in the Self-similarity section is wrong.

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contains all numbers which do not have a 1 digit on first two fraction places, plus {0.01, 0.1, 0.21}, which

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Yes. Every nonempty totally-disconnected perfect compact metric space is homeomorphic to the Cantor set. --

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Even though I know why the space in question is homeomorphic to the Cantor set, the sentence quoted does

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Well, I was wrong. The sequence of sets in the article is not monotone, and the formula is really wrong.

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The Cantor set is not countably infinite. It has the same cardinality as the set of real numbers. — Carl

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The Cantor set is compact, the set of all irrationals is not compact. So they cannot be homeomorphic. --

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Which crucially states that the union of the left and right transformations is precisely the Cantor set.

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Its also incorrect to say that the "intervals shrink to points" -- they don't, they remain intervals.

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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.

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I agree with you that a proof without AC, where it can be avoided, is more elegant than one with it.

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There should be a section here, noting that some deny the existence of the Cantor set altogether.

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One can see that Γ is totally disconnected and perfect - thus it is homeomorphic to the Cantor set.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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countable, so there are uncountably many non-repeating ternaries consisting only of 0s and 2s. --

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belongs to the Cantor set. Its ternary contains only 0s and 2s, but neither the 0 nor the 2 is

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groups you only get rational numbers, and you know those are countable. But the Cantor set is

3653:{\displaystyle 1/4=2\cdot 3^{-2}+2\cdot 3^{-4}+2\cdot 3^{-6}+\cdots =0{,}{\overline {02}}_{3}} 3058: 2374: 1990: 1230: 725: 698:

Prose was "you do this", "if you add up" and the like. I hope my rewrite is more encyclopædic

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That is not the Cantor set (the direct sum is only countable), so: What is the connection?

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I personally feel that the AC should be avoided whenever possible. What do you guys think?

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Cutting off the first middle b-2/b in the first iteration leaves us with intervals and

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If so, why are we concerned with the "countable direct sum" of this group with itself???

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is it appropriate to add that it's a perfect subset of R, uncountable, it is a zero set ?

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is to the right of the interval that was removed (in the n step of the construction of

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Because of finite size of the real-world fractal structures, the correct model is the

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If you think it's countably infinite, you haven't read the article carefully enough.

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Nowhere dense means not dense in any open set. Every set is dense in itself. — Carl

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uses a different disallowed digit, then there's an infinite number of decimal sets.

647:"particular" implies that he had a specific series in mind, which I seriously doubt. 3362:

At layer 0 the Cantor set. (The set of numbers which does not contain the digit 1.)

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contains all numbers but those with 1 at 1-st place — except 0.1 which belongs to C

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For nowhere dense, the property proven doesn't quite align with the definition in

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Firstly, I'm just a boy interested in maths, so bear with me if i am wrong. But:

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I think the first explicit formula of the set is wrong. I think that instead of

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I was wrong, the equation reads 1/3 * 1/(1/3) which is equal to one. Oops. --

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If you agree, I think someone should change it! Otherwise, please correct me.

2934:). Of course, once this theorem is proved, it doesn't need to be proved again. 3765:

There are an intuitive recursive construction rule (illustrated) for each new

2202:). I'm pretty sure that it is not necessary. For example, a bijection between 1371: 1296:

Rough fix, markup looks a little weird but at least the equation is right. --

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It's not finite and it's not countably infinite; it's uncountably infinite.

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th iteration)." These points, clearly, are all rational numbers--but while

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of the integers, or equivalently of the power set of the positive integers.

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The article stated that "the remaining points are all numbers of the form

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in a different way... Instead of intersection the hierarchy is an union:

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The Cantor cubes only show about six stages, six being a finite number.

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I'm wondering if any of this would be useful to add to the encyclopedia.

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Corrected the reference to Cantor's original paper in Acta Mathematica.

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The Cantor-Bernstein-Schroeder theorem is the simplest way to take the

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it should be removed or the reference should be moved somewhere else.

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The thing about the Cantor set in mathematics is that the term is used

3500:" Its ternary contains only 0s and 2s, but neither the 0 nor the 2 is 2063: 969: 903:=, that is numbers beginning whith zero: 0.... plus the ending one: 1. 720:

added a note that the "middle 80%" version works nicely with decimals

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The intervals created will be 1/n of the size of the previous ones.

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Knowledge:Redirects for discussion/Log/2022 October 15#Cantor`s dust

3382:(Does this set have a name? What are its topological properties?) - 2281:{\displaystyle f(x):=(.\alpha _{1}\alpha _{2}\alpha _{3}\dots )_{2}} 4168:

are encyclopedic. It is important to introduce it in the article.

1501:{\displaystyle f_{L}(C)\cong f_{R}(C)\cong C=f_{L}(C)\cup f_{R}(C)} 4357: 4314: 4274: 4177: 3945: 3683: 3519: 3460: 3435: 3410: 3391: 3344: 3250: 3214: 3157: 3143: 3048: 2943: 2805: 2771: 2378: 2057: 2030: 1998: 1981: 1917: 1896: 1880: 1860: 1807: 1661: 1519: 1387: 1359: 1341: 1319: 1305: 1269: 1254: 1238: 1203: 1154: 1119: 1093: 1074: 1049: 1018: 989: 978: 882: 855: 835: 825: 777: 767: 749: 745:

says the cantor set is a null set. This should be notied if true.

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is used for more than one thing in mathematics, the meaning of Z

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In the proof that the Cantor Set is uncountable, it says that:

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and the set of infinite binary sequences could be constructed by

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An editor has identified a potential problem with the redirect

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but I think that what was actually intended was something like

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and for that reason it is unclear what the meaning of "is" is.

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I'm not sure what proof you're referring to, though, because

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Cantor set" the set defined by "removing the middle third".

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I don't think they do the "middle third" contruction either.

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I just thought it was cool to note that in this statement:

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is the set of all of 2 numbers expressed as fixed-length (

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The following is just as true, ambiguous, and whimsical:

2755:{\displaystyle 2^{\aleph _{0}}=\aleph _{2^{\aleph _{0}}}} 3758:

is the domain of a set of identifiers (or indexes), and

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fits into the sequence of cardinal numbers. (It must be

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The trigonometric-series origins of Cantor's set theory

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totally disconnected, dense-in-itself, and metrizable.

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The answer is that the options a and b are equivalent

4211:(the countable direct sum) is discrete. Although the 4059: 3998: 3538: 3371:

At layer 3 nine copies. (Last digit 1 at position 3.)

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http://swiki.hfbk-hamburg.de:8888/MusicTechnology/799

799:/3 does not lie in any middle third (in other words, 292:

Intro defines Cantor set incorrectly (intro is wrong)

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For any integer q ≥ 2, the topology on the group G=Z

112:, a collaborative effort to improve the coverage of 3304:

is the q-adic integers (the only other meaning of Z

525:. Should they be merged to reduce the confusion? – 4149: 4045: 3652: 3228:contains this passage (with the formula omitted): 3074: 3023: 2982: 2926: 2906: 2886: 2860: 2834: 2786:equal, by the Cantor–Bernstein–Schroeder theorem. 2754: 2693: 2653: 2633: 2603: 2583: 2556: 2502: 2478: 2428: 2401: 2358: 2334: 2314: 2280: 2194: 2136: 2116: 1949:with the digits as is done in the presented proof 1787: 1723: 1644: 1580: 1500: 895:denote the n-th approximation of the Cantor set. 4150:{\displaystyle |T_{k}|=|T_{k-1}|+2^{k}=2^{k+1}-2} 3470:repeating fraction with a repetend that uses only 1276:"What's in the Cantor Set?" sum equation is wrong 1005:Well, the Cantor set clearly contais 1 and 0. !!! 3063: 2965: 866:explain how these "extra" points made it into... 4392:Knowledge level-5 vital articles in Mathematics 3134:) equal to 1 on the "limit set" and 0 outside. 1667:You're raising a good point. You're right that 755:The middle-third construction gives a set with 680:How about "discovered" rather than "invented"? 617:)? What's your reference for this information? 3951:Please add hierarchical tree of the Cantor set 3689:Please explain the mapping to binary sequences 936:contains all numbers, which do not have digit 4046:{\displaystyle T_{k}=\bigcup _{m=1}^{k}C_{m}} 3731:= {0, 00, 000, 001, 01, 010, 011, ..., 111}; 2409:by itself doesn't have any standard meaning. 1927:In the paragraph "Cardinality" it is stated: 1788:{\displaystyle f_{L}(C)\cong f_{R}(C)\cong C} 495:just the other way around to mathematicians. 8: 842:To answer my own question: 1/4 = 0.020202... 624:Since you didn't respond, I'm erasing this. 217:, which collaborates on articles related to 4244:denotes the group of integers modulo q ??? 3226:Construction and formula of the ternary set 2195:{\displaystyle \Rightarrow |C|\geq \aleph } 4187:The next-to-last paragraph in the section 4166:hierarchical tree of the finite Cantor set 3446: 3421: 967:decimal, or with an algebraic expression. 163: 58: 4129: 4116: 4104: 4092: 4083: 4075: 4069: 4060: 4058: 4037: 4027: 4016: 4003: 3997: 3644: 3634: 3628: 3607: 3585: 3563: 3542: 3537: 3066: 3060: 3024:{\displaystyle \bigcap _{m=1}^{\infty }.} 3012: 3001: 2995: 2968: 2962: 2919: 2899: 2873: 2847: 2821: 2742: 2737: 2732: 2717: 2712: 2706: 2683: 2678: 2666: 2646: 2620: 2596: 2575: 2569: 2546: 2541: 2529: 2494: 2493: 2491: 2468: 2463: 2451: 2443: 2441: 2420: 2414: 2394: 2351: 2327: 2300: 2294: 2272: 2259: 2249: 2239: 2212: 2181: 2173: 2153: 2129: 2124:uses the axiom of choice (by saying that 2103: 2095: 2093: 1764: 1742: 1736: 1700: 1678: 1672: 1621: 1599: 1593: 1557: 1535: 1529: 1483: 1461: 1433: 1411: 1405: 517:Note: Knowledge has two articles, namely 3314:: What does it *mean* where it states: 2084:Proof regarding cardinality & the AC 4382:Knowledge vital articles in Mathematics 4338:This can be written in base b as and 4262:instead denotes the q-adic integers??? 3512:2604:2000:81CF:2B00:A1AD:998B:217B:9B39 3350:Interesting extension of the Cantor set 3327:??? Whatever the intended meaning of Z 1724:{\displaystyle f_{L}(C)\cup f_{R}(C)=C} 1645:{\displaystyle f_{L}(C)\cup f_{R}(C)=C} 165: 60: 19: 3453:2A00:23C0:FCF6:4801:C43C:4F7C:53E0:BB3 3428:2A00:23C0:FCF6:4801:C43C:4F7C:53E0:BB3 3300:If on the other hand, the meaning of Z 2088:The current proof in the article that 1032:" Should this be removed or updated? 4397:C-Class vital articles in Mathematics 4189:Topological and analytical properties 3262:Topological and analytical properties 3172:Topological and analytical properties 2694:{\displaystyle \psi =2^{\aleph _{0}}} 7: 3397:Suggestions for "Properties" section 2983:{\displaystyle \lim _{m\to \infty }} 996:The Cantor set contains no intervals 783:Irrational numbers in the Cantor set 311:Cantor himself defined his set as a 273:This article is within the field of 211:This article is within the scope of 106:This article is within the scope of 2861:{\displaystyle \delta \leq \gamma } 2835:{\displaystyle \gamma \leq \delta } 2479:{\displaystyle |C|=2^{\aleph _{0}}} 1581:{\displaystyle f_{L}(C)=f_{R}(C)=C} 49:It is of interest to the following 4320:Generalization of a mentioned fact 3260:The last paragraph in the section 3013: 2975: 2739: 2729: 2714: 2680: 2572: 2543: 2465: 2417: 2396: 2189: 2111: 1125: 14: 4407:Mid-priority mathematics articles 4200:≥ 2, the topology on the group G= 2634:{\displaystyle \psi \neq \omega } 2557:{\displaystyle c=2^{\aleph _{0}}} 126:Knowledge:WikiProject Mathematics 4422:Systems articles in chaos theory 4377:Knowledge level-5 vital articles 4348:base b with only 0s and (b-1)s. 4300:. This discussion will occur at 4287: 3694: 3198:homeomorphic to the Cantor set. 1055:Method to generate arbitary sets 968: 940:in their ternary representation 925:with a 1 at one of those places. 899:Initially, we have an interval C 198: 188: 167: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 4417:Mid-importance Systems articles 3319:the Pontrjagin dual Γ is also Z 3277:, the topology of Γ is compact. 2887:{\displaystyle \gamma =\delta } 2584:{\displaystyle \aleph _{\psi }} 1040:Fractals with continous curves? 955:0(dot)* | 0(dot)*10* | 1(dot)0* 659:Michael, are you OK with this? 251:This article has been rated as 146:This article has been rated as 4387:C-Class level-5 vital articles 4105: 4084: 4076: 4061: 3927:PS: by induction we see that | 3751:In usual applications the set 2972: 2944:19:21, 24 September 2014 (UTC) 2772:19:14, 24 September 2014 (UTC) 2503:{\displaystyle {\mathcal {P}}} 2452: 2444: 2269: 2229: 2223: 2217: 2182: 2174: 2170: 2167: 2155: 2104: 2096: 1837:) 12:24, 14 January 2009 (UTC) 1776: 1770: 1754: 1748: 1712: 1706: 1690: 1684: 1633: 1627: 1611: 1605: 1569: 1563: 1547: 1541: 1495: 1489: 1473: 1467: 1445: 1439: 1423: 1417: 512:23:38, 27 September 2014 (UTC) 456:03:22, 17 September 2007 (UTC) 438:02:40, 17 September 2017 (UTC) 1: 4265:Clarification is needed here. 3840:= {00, 01, 10, 11} U {0, 1}. 3497:II. From your comment below: 3474:I. The sentence in question: 3251:03:07, 7 September 2014 (UTC) 3215:02:49, 7 September 2014 (UTC) 3091:in the sequence the function 2510:(ℤ)|, the cardinality of the 2486:. :Put another way, |C| = | 2315:{\displaystyle \alpha _{n}=1} 2031:20:04, 26 December 2009 (UTC) 1999:10:39, 23 December 2009 (UTC) 1982:15:28, 11 December 2009 (UTC) 1520:16:18, 28 February 2018 (UTC) 1219:20:55, 10 December 2007 (UTC) 1204:20:28, 10 December 2007 (UTC) 1186:19:50, 10 December 2007 (UTC) 1171:16:28, 10 December 2007 (UTC) 1155:16:22, 10 December 2007 (UTC) 1140:14:55, 10 December 2007 (UTC) 1120:20:59, 12 November 2007 (UTC) 1094:13:18, 12 November 2007 (UTC) 1075:11:46, 12 November 2007 (UTC) 1050:11:31, 12 November 2007 (UTC) 231:Knowledge:WikiProject Systems 120:and see a list of open tasks. 4427:WikiProject Systems articles 4402:C-Class mathematics articles 4358:13:10, 9 February 2024 (UTC) 4315:15:28, 15 October 2022 (UTC) 4275:15:14, 28 October 2020 (UTC) 3960:produced by light) see e.g. 3684:19:59, 20 January 2019 (UTC) 3639: 3520:13:01, 21 January 2019 (UTC) 3392:20:56, 24 October 2017 (UTC) 3290:here needs to be specified. 2078:03:24, 7 February 2010 (UTC) 2058:20:45, 23 October 2017 (UTC) 1897:07:03, 28 October 2017 (UTC) 1861:12:30, 14 January 2009 (UTC) 1808:23:59, 14 October 2008 (UTC) 1662:20:09, 14 October 2008 (UTC) 1388:06:57, 28 October 2017 (UTC) 1360:19:54, 12 October 2008 (UTC) 1342:19:42, 12 October 2008 (UTC) 1270:21:16, 23 October 2017 (UTC) 1255:21:16, 23 October 2017 (UTC) 1224:Variations of the Cantor Set 1019:21:23, 16 October 2006 (UTC) 778:20:07, 1 November 2006 (UTC) 480:15:56, 18 October 2011 (UTC) 234:Template:WikiProject Systems 3122:) converges for every real 2429:{\displaystyle \aleph _{0}} 2117:{\displaystyle |C|=\aleph } 639:I edited it again, because 460:Missing the Significance? 4443: 4280:"Cantor`s dust" listed at 4230:=2. (See Rudin 1962 p 40.) 4178:16:55, 24 March 2019 (UTC) 3946:11:04, 17 March 2019 (UTC) 3722:= {0, 00, 01, 1, 10, 11}; 2701:, or in other words that 2611:that must satisfy 1 <= 1320:15:55, 28 April 2008 (UTC) 1306:08:57, 27 April 2008 (UTC) 1291:16:04, 25 April 2008 (UTC) 990:19:37, 23 March 2007 (UTC) 979:19:29, 21 April 2006 (UTC) 883:20:05, 30 March 2006 (UTC) 856:18:56, 30 March 2006 (UTC) 836:00:39, 29 March 2006 (UTC) 826:18:18, 28 March 2006 (UTC) 795:is a positive integer and 689:Notes on Rewrite, May 2005 607: 603:17:56, 18 April 2006 (UTC) 257:project's importance scale 3844:The number of elements, | 3345:16:42, 1 March 2016 (UTC) 3166:Massively false statement 3075:{\displaystyle \lim _{n}} 2806:22:35, 10 July 2012 (UTC) 2379:20:34, 4 March 2010 (UTC) 1881:03:27, 4 March 2009 (UTC) 1332:Are 0 and 1 in this set? 1239:01:06, 8 March 2008 (UTC) 1126:What's In The Cantor Set? 768:22:59, 3 March 2006 (UTC) 761:Smith-Volterra-Cantor set 750:22:02, 3 March 2006 (UTC) 621:10:55, 29 Jul 2004 (UTC) 589:07:02, 15 Jul 2004 (UTC) 574:01:07, 27 Nov 2003 (UTC) 535:08:15, 24 July 2015 (UTC) 404:08:31, 6 March 2006 (UTC) 378:20:23, 5 March 2006 (UTC) 357:22:53, 3 March 2006 (UTC) 340:22:04, 3 March 2006 (UTC) 328:05:03, 26 July 2005 (UTC) 272: 250: 183: 145: 78: 57: 4412:C-Class Systems articles 4282:Redirects for discussion 4053:. We can show also that 3916:| = 14+16=30; ...; | 3816:) bit strings. Example: 3461:07:07, 5 June 2018 (UTC) 3436:07:15, 4 June 2018 (UTC) 3282:But since the notation Z 3220:Statement makes no sense 3174:contains this statment: 3158:23:23, 25 May 2013 (UTC) 3144:22:50, 25 May 2013 (UTC) 3049:20:04, 25 May 2013 (UTC) 1918:21:30, 2 July 2009 (UTC) 932:Finally the Cantor set C 684:21:36, 24 Sep 2004 (UTC) 671:15:02, 16 Aug 2004 (UTC) 663:07:03, 4 Aug 2004 (UTC) 636:16:03, 3 Aug 2004 (UTC) 628:07:26, 3 Aug 2004 (UTC) 152:project's priority scale 4296:and has thus listed it 3858:|=2, is the recurrence 3411:19:33, 2 May 2018 (UTC) 3264:contains this passage: 3109:and to 0 outside, then 3055:You are not wrong, but 2927:{\displaystyle \delta } 2907:{\displaystyle \gamma } 2816:of the statements that 2654:{\displaystyle \omega } 2402:{\displaystyle \aleph } 729:08:40, 4 May 2005 (UTC) 578:Removed textual picture 109:WikiProject Mathematics 4372:C-Class vital articles 4151: 4047: 4032: 3654: 3527:No, you are not right. 3184:This is very false. 3100:equal to 1 on the set 3076: 3025: 3017: 2984: 2928: 2908: 2888: 2862: 2836: 2778:Reference Unnecessary? 2756: 2695: 2655: 2635: 2605: 2585: 2558: 2504: 2480: 2430: 2403: 2360: 2336: 2316: 2282: 2196: 2138: 2118: 1903:Practical applications 1789: 1725: 1646: 1582: 1502: 269: 206:Systems science portal 4152: 4048: 4012: 3891:ranging from 2 to 8: 3655: 3077: 3026: 2997: 2985: 2929: 2909: 2889: 2868:and to conclude that 2863: 2837: 2757: 2696: 2656: 2636: 2606: 2604:{\displaystyle \psi } 2586: 2559: 2505: 2481: 2431: 2404: 2361: 2337: 2317: 2283: 2197: 2139: 2119: 1790: 1726: 1647: 1583: 1503: 1024:This link is obsolete 486:The thing about it is 268: 36:level-5 vital article 4057: 3996: 3985:use the constructor 3536: 3256:Confusing statements 3059: 2994: 2990:, there should be a 2961: 2918: 2898: 2872: 2846: 2820: 2814:mathematical meaning 2705: 2665: 2645: 2619: 2595: 2568: 2528: 2490: 2440: 2413: 2393: 2350: 2346:) from the interval 2326: 2293: 2211: 2152: 2128: 2092: 1923:Unsatisfactory proof 1735: 1671: 1592: 1528: 1404: 847:0.220200200020000... 132:mathematics articles 4235:Can we assume that 3190:Another example is 1036:, 18 November 2006 214:WikiProject Systems 4191:reads as follows: 4160:All concepts, the 4147: 4043: 3650: 3072: 3071: 3021: 2980: 2979: 2924: 2904: 2884: 2858: 2832: 2752: 2691: 2651: 2631: 2601: 2581: 2554: 2500: 2476: 2426: 2399: 2356: 2332: 2312: 2278: 2192: 2134: 2114: 1814:Irrational numbers 1785: 1721: 1642: 1578: 1498: 270: 101:Mathematics portal 45:content assessment 4183:Unclear paragraph 4162:finite Cantor set 3973:finite Cantor set 3745: 3744: 3642: 3463: 3451:comment added by 3438: 3426:comment added by 3333:topological group 3062: 3039:comment added by 2964: 2796:comment added by 2591:for some ordinal 2359:{\displaystyle x} 2335:{\displaystyle x} 2137:{\displaystyle f} 2021:comment added by 1972:comment added by 1879: 1864: 1847:comment added by 1839: 1825:comment added by 1806: 1358: 1202: 1122: 1106:comment added by 1092: 653:set of uniqueness 615:set of uniqueness 470:comment added by 303:). One may call " 289: 288: 285: 284: 281: 280: 162: 161: 158: 157: 4434: 4291: 4196:For any integer 4156: 4154: 4153: 4148: 4140: 4139: 4121: 4120: 4108: 4103: 4102: 4087: 4079: 4074: 4073: 4064: 4052: 4050: 4049: 4044: 4042: 4041: 4031: 4026: 4008: 4007: 3923:| = 254+256=510. 3705: 3704: 3698: 3659: 3657: 3656: 3651: 3649: 3648: 3643: 3635: 3632: 3615: 3614: 3593: 3592: 3571: 3570: 3546: 3358:What we get is: 3081: 3079: 3078: 3073: 3070: 3051: 3030: 3028: 3027: 3022: 3016: 3011: 2989: 2987: 2986: 2981: 2978: 2950:Explicit formula 2933: 2931: 2930: 2925: 2913: 2911: 2910: 2905: 2893: 2891: 2890: 2885: 2867: 2865: 2864: 2859: 2841: 2839: 2838: 2833: 2808: 2761: 2759: 2758: 2753: 2751: 2750: 2749: 2748: 2747: 2746: 2724: 2723: 2722: 2721: 2700: 2698: 2697: 2692: 2690: 2689: 2688: 2687: 2660: 2658: 2657: 2652: 2640: 2638: 2637: 2632: 2610: 2608: 2607: 2602: 2590: 2588: 2587: 2582: 2580: 2579: 2563: 2561: 2560: 2555: 2553: 2552: 2551: 2550: 2509: 2507: 2506: 2501: 2499: 2498: 2485: 2483: 2482: 2477: 2475: 2474: 2473: 2472: 2455: 2447: 2435: 2433: 2432: 2427: 2425: 2424: 2408: 2406: 2405: 2400: 2365: 2363: 2362: 2357: 2341: 2339: 2338: 2333: 2321: 2319: 2318: 2313: 2305: 2304: 2287: 2285: 2284: 2279: 2277: 2276: 2264: 2263: 2254: 2253: 2244: 2243: 2201: 2199: 2198: 2193: 2185: 2177: 2143: 2141: 2140: 2135: 2123: 2121: 2120: 2115: 2107: 2099: 2033: 1984: 1869: 1863: 1841: 1838: 1819: 1796: 1794: 1792: 1791: 1786: 1769: 1768: 1747: 1746: 1730: 1728: 1727: 1722: 1705: 1704: 1683: 1682: 1651: 1649: 1648: 1643: 1626: 1625: 1604: 1603: 1587: 1585: 1584: 1579: 1562: 1561: 1540: 1539: 1507: 1505: 1504: 1499: 1488: 1487: 1466: 1465: 1438: 1437: 1416: 1415: 1348: 1192: 1101: 1082: 972: 527:Tobias Bergemann 482: 401:Tobias Bergemann 239: 238: 237:Systems articles 235: 232: 229: 208: 203: 202: 201: 192: 185: 184: 179: 171: 164: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 4442: 4441: 4437: 4436: 4435: 4433: 4432: 4431: 4362: 4361: 4322: 4285: 4261: 4243: 4224: 4213:Pontrjagin dual 4209: 4185: 4125: 4112: 4088: 4065: 4055: 4054: 4033: 3999: 3994: 3993: 3991: 3984: 3953: 3933: 3922: 3915: 3909:| = 6+8=14; | 3908: 3901: 3886: 3875: 3868: 3857: 3850: 3839: 3832: 3825: 3811: 3801: 3794: 3787: 3778: 3771: 3757: 3739: 3730: 3721: 3713: 3691: 3633: 3603: 3581: 3559: 3534: 3533: 3472: 3418: 3399: 3352: 3330: 3322: 3307: 3303: 3289: 3285: 3276: 3272: 3258: 3222: 3168: 3117: 3108: 3099: 3090: 3057: 3056: 3034: 2992: 2991: 2959: 2958: 2952: 2916: 2915: 2896: 2895: 2894:(for cardinals 2870: 2869: 2844: 2843: 2818: 2817: 2791: 2780: 2738: 2733: 2728: 2713: 2708: 2703: 2702: 2679: 2674: 2663: 2662: 2643: 2642: 2617: 2616: 2593: 2592: 2571: 2566: 2565: 2542: 2537: 2526: 2525: 2488: 2487: 2464: 2459: 2438: 2437: 2416: 2411: 2410: 2391: 2390: 2348: 2347: 2324: 2323: 2296: 2291: 2290: 2268: 2255: 2245: 2235: 2209: 2208: 2150: 2149: 2126: 2125: 2090: 2089: 2086: 2066: 2016: 1967: 1943: 1941: 1937: 1933: 1925: 1908:left-brainers. 1905: 1842: 1820: 1816: 1760: 1738: 1733: 1732: 1696: 1674: 1669: 1668: 1617: 1595: 1590: 1589: 1553: 1531: 1526: 1525: 1479: 1457: 1429: 1407: 1402: 1401: 1398: 1396:Self-similarity 1330: 1328:Layman question 1278: 1226: 1128: 1057: 1042: 1026: 1012: 998: 935: 920: 913: 909: 902: 894: 850: 845: 785: 739: 691: 610: 595: 584: 580: 560: 544: 465: 294: 236: 233: 230: 227: 226: 223:systems science 204: 199: 197: 177: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 4440: 4438: 4430: 4429: 4424: 4419: 4414: 4409: 4404: 4399: 4394: 4389: 4384: 4379: 4374: 4364: 4363: 4321: 4318: 4298:for discussion 4284: 4278: 4257: 4239: 4220: 4205: 4184: 4181: 4146: 4143: 4138: 4135: 4132: 4128: 4124: 4119: 4115: 4111: 4107: 4101: 4098: 4095: 4091: 4086: 4082: 4078: 4072: 4068: 4063: 4040: 4036: 4030: 4025: 4022: 4019: 4015: 4011: 4006: 4002: 3989: 3982: 3952: 3949: 3931: 3925: 3924: 3920: 3913: 3906: 3902:| = 2+4=6; | 3899: 3884: 3878: 3877: 3873: 3866: 3855: 3848: 3842: 3841: 3837: 3830: 3823: 3809: 3803: 3802: 3799: 3792: 3785: 3776: 3769: 3755: 3749: 3748: 3747: 3746: 3743: 3742: 3737: 3728: 3719: 3711: 3690: 3687: 3665: 3647: 3641: 3638: 3631: 3627: 3624: 3621: 3618: 3613: 3610: 3606: 3602: 3599: 3596: 3591: 3588: 3584: 3580: 3577: 3574: 3569: 3566: 3562: 3558: 3555: 3552: 3549: 3545: 3541: 3528: 3496: 3471: 3468: 3467: 3466: 3465: 3464: 3417: 3414: 3398: 3395: 3376: 3375: 3374:At layer 4 ... 3372: 3369: 3366: 3363: 3351: 3348: 3328: 3320: 3305: 3301: 3287: 3283: 3274: 3270: 3257: 3254: 3221: 3218: 3167: 3164: 3163: 3162: 3161: 3160: 3113: 3104: 3095: 3086: 3069: 3065: 3041:188.183.128.99 3020: 3015: 3010: 3007: 3004: 3000: 2977: 2974: 2971: 2967: 2951: 2948: 2947: 2946: 2923: 2903: 2883: 2880: 2877: 2857: 2854: 2851: 2831: 2828: 2825: 2779: 2776: 2775: 2774: 2745: 2741: 2736: 2731: 2727: 2720: 2716: 2711: 2686: 2682: 2677: 2673: 2670: 2650: 2630: 2627: 2624: 2615:<= |C| and 2600: 2578: 2574: 2549: 2545: 2540: 2536: 2533: 2522:direct product 2516: 2515: 2497: 2471: 2467: 2462: 2458: 2454: 2450: 2446: 2423: 2419: 2398: 2386: 2385: 2368: 2367: 2355: 2331: 2311: 2308: 2303: 2299: 2288: 2275: 2271: 2267: 2262: 2258: 2252: 2248: 2242: 2238: 2234: 2231: 2228: 2225: 2222: 2219: 2216: 2207: 2191: 2188: 2184: 2180: 2176: 2172: 2169: 2166: 2163: 2160: 2157: 2133: 2113: 2110: 2106: 2102: 2098: 2085: 2082: 2065: 2062: 2061: 2060: 2037: 2036: 2035: 2034: 2012: 2008: 2002: 2001: 1974:213.240.234.31 1939: 1935: 1931: 1929: 1924: 1921: 1904: 1901: 1900: 1899: 1884: 1883: 1815: 1812: 1811: 1810: 1784: 1781: 1778: 1775: 1772: 1767: 1763: 1759: 1756: 1753: 1750: 1745: 1741: 1720: 1717: 1714: 1711: 1708: 1703: 1699: 1695: 1692: 1689: 1686: 1681: 1677: 1641: 1638: 1635: 1632: 1629: 1624: 1620: 1616: 1613: 1610: 1607: 1602: 1598: 1577: 1574: 1571: 1568: 1565: 1560: 1556: 1552: 1549: 1546: 1543: 1538: 1534: 1497: 1494: 1491: 1486: 1482: 1478: 1475: 1472: 1469: 1464: 1460: 1456: 1453: 1450: 1447: 1444: 1441: 1436: 1432: 1428: 1425: 1422: 1419: 1414: 1410: 1397: 1394: 1393: 1392: 1391: 1390: 1363: 1362: 1329: 1326: 1325: 1324: 1323: 1322: 1277: 1274: 1273: 1272: 1257: 1225: 1222: 1207: 1206: 1158: 1157: 1127: 1124: 1097: 1096: 1056: 1053: 1041: 1038: 1025: 1022: 1011: 1008: 1007: 1006: 997: 994: 993: 992: 957: 956: 933: 930: 929: 926: 918: 915: 911: 907: 904: 900: 892: 890: 888: 887: 886: 885: 872: 871: 870: 869: 859: 858: 848: 843: 839: 838: 784: 781: 771: 770: 738: 735: 733: 722: 721: 718: 715: 708: 705: 702: 699: 690: 687: 686: 685: 673: 672: 657: 656: 648: 645: 609: 606: 594: 591: 581: 579: 576: 559: 556: 554:, 2001 Nov 30 543: 540: 539: 538: 453:116.197.236.12 449: 448: 447: 446: 445: 444: 443: 442: 441: 440: 415: 413: 412: 411: 410: 409: 408: 407: 406: 387: 386: 385: 384: 383: 382: 381: 380: 364: 363: 362: 361: 360: 359: 345: 344: 343: 342: 310: 293: 290: 287: 286: 283: 282: 279: 278: 271: 261: 260: 253:Mid-importance 249: 243: 242: 240: 210: 209: 193: 181: 180: 178:Mid‑importance 172: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 4439: 4428: 4425: 4423: 4420: 4418: 4415: 4413: 4410: 4408: 4405: 4403: 4400: 4398: 4395: 4393: 4390: 4388: 4385: 4383: 4380: 4378: 4375: 4373: 4370: 4369: 4367: 4360: 4359: 4355: 4351: 4350:AbrarBinCiraj 4345: 4342: 4339: 4336: 4333: 4330: 4326: 4319: 4317: 4316: 4313: 4310: 4307: 4303: 4299: 4295: 4294:Cantor`s dust 4290: 4283: 4279: 4277: 4276: 4272: 4268: 4263: 4260: 4256: 4251: 4248: 4245: 4242: 4238: 4233: 4231: 4227: 4223: 4219: 4218: 4214: 4208: 4204: 4203: 4197: 4192: 4190: 4182: 4180: 4179: 4175: 4171: 4167: 4163: 4158: 4144: 4141: 4136: 4133: 4130: 4126: 4122: 4117: 4113: 4109: 4099: 4096: 4093: 4089: 4080: 4070: 4066: 4038: 4034: 4028: 4023: 4020: 4017: 4013: 4009: 4004: 4000: 3988: 3981: 3976: 3974: 3969: 3967: 3963: 3959: 3950: 3948: 3947: 3943: 3939: 3935: 3930: 3919: 3912: 3905: 3898: 3894: 3893: 3892: 3890: 3883: 3880:Examples of | 3872: 3865: 3861: 3860: 3859: 3854: 3847: 3836: 3829: 3822: 3819: 3818: 3817: 3815: 3808: 3798: 3791: 3784: 3781: 3780: 3779: 3775: 3768: 3763: 3761: 3754: 3741: 3736: 3732: 3727: 3723: 3718: 3710: 3707: 3706: 3703: 3702: 3701: 3700: 3699: 3697: 3688: 3686: 3685: 3681: 3677: 3673: 3669: 3663: 3645: 3636: 3629: 3625: 3622: 3619: 3616: 3611: 3608: 3604: 3600: 3597: 3594: 3589: 3586: 3582: 3578: 3575: 3572: 3567: 3564: 3560: 3556: 3553: 3550: 3547: 3543: 3539: 3531: 3525: 3521: 3517: 3513: 3509: 3505: 3503: 3498: 3494: 3491: 3486: 3483: 3479: 3475: 3469: 3462: 3458: 3454: 3450: 3444: 3443: 3441: 3440: 3439: 3437: 3433: 3429: 3425: 3415: 3413: 3412: 3408: 3404: 3396: 3394: 3393: 3389: 3385: 3380: 3373: 3370: 3367: 3364: 3361: 3360: 3359: 3356: 3349: 3347: 3346: 3342: 3338: 3334: 3325: 3323: 3315: 3313: 3309: 3298: 3296: 3291: 3280: 3278: 3265: 3263: 3255: 3253: 3252: 3248: 3244: 3239: 3236: 3234: 3229: 3227: 3219: 3217: 3216: 3212: 3208: 3204: 3199: 3197: 3193: 3188: 3185: 3182: 3180: 3175: 3173: 3165: 3159: 3155: 3151: 3147: 3146: 3145: 3141: 3137: 3133: 3129: 3125: 3121: 3116: 3112: 3107: 3103: 3098: 3094: 3089: 3085: 3067: 3054: 3053: 3052: 3050: 3046: 3042: 3038: 3031: 3018: 3008: 3005: 3002: 2998: 2969: 2955: 2949: 2945: 2941: 2937: 2921: 2901: 2881: 2878: 2875: 2855: 2852: 2849: 2829: 2826: 2823: 2815: 2811: 2810: 2809: 2807: 2803: 2799: 2795: 2787: 2783: 2777: 2773: 2769: 2765: 2743: 2734: 2725: 2718: 2709: 2684: 2675: 2671: 2668: 2648: 2628: 2625: 2622: 2614: 2598: 2576: 2547: 2538: 2534: 2531: 2523: 2518: 2517: 2513: 2469: 2460: 2456: 2448: 2421: 2388: 2387: 2383: 2382: 2381: 2380: 2376: 2372: 2353: 2345: 2329: 2309: 2306: 2301: 2297: 2273: 2265: 2260: 2256: 2250: 2246: 2240: 2236: 2232: 2226: 2220: 2214: 2205: 2186: 2178: 2164: 2161: 2158: 2147: 2131: 2108: 2100: 2083: 2081: 2079: 2075: 2071: 2059: 2055: 2051: 2047: 2043: 2042:by definition 2039: 2038: 2032: 2028: 2024: 2023:87.121.84.214 2020: 2013: 2009: 2006: 2005: 2004: 2003: 2000: 1996: 1992: 1987: 1986: 1985: 1983: 1979: 1975: 1971: 1965: 1961: 1958: 1954: 1950: 1946: 1938:= 0.022222... 1928: 1922: 1920: 1919: 1915: 1911: 1902: 1898: 1894: 1890: 1886: 1885: 1882: 1877: 1873: 1867: 1866: 1865: 1862: 1858: 1854: 1850: 1846: 1836: 1832: 1828: 1824: 1813: 1809: 1804: 1800: 1782: 1779: 1773: 1765: 1761: 1757: 1751: 1743: 1739: 1718: 1715: 1709: 1701: 1697: 1693: 1687: 1679: 1675: 1666: 1665: 1664: 1663: 1659: 1655: 1639: 1636: 1630: 1622: 1618: 1614: 1608: 1600: 1596: 1575: 1572: 1566: 1558: 1554: 1550: 1544: 1536: 1532: 1522: 1521: 1517: 1513: 1508: 1492: 1484: 1480: 1476: 1470: 1462: 1458: 1454: 1451: 1448: 1442: 1434: 1430: 1426: 1420: 1412: 1408: 1395: 1389: 1385: 1381: 1377: 1373: 1369: 1368: 1365: 1364: 1361: 1356: 1352: 1346: 1345: 1344: 1343: 1339: 1335: 1334:118.90.35.242 1327: 1321: 1317: 1313: 1309: 1308: 1307: 1303: 1299: 1295: 1294: 1293: 1292: 1288: 1284: 1271: 1267: 1263: 1258: 1256: 1252: 1248: 1243: 1242: 1241: 1240: 1236: 1232: 1223: 1221: 1220: 1216: 1212: 1211:Michael Hardy 1205: 1200: 1196: 1190: 1189: 1188: 1187: 1183: 1179: 1178:Martin Packer 1173: 1172: 1168: 1164: 1163:Michael Hardy 1156: 1152: 1148: 1147:Michael Hardy 1144: 1143: 1142: 1141: 1137: 1133: 1132:Martin Packer 1123: 1121: 1117: 1113: 1109: 1105: 1095: 1090: 1086: 1079: 1078: 1077: 1076: 1073: 1068: 1064: 1060: 1054: 1052: 1051: 1048: 1039: 1037: 1035: 1031: 1023: 1021: 1020: 1017: 1009: 1004: 1003: 1002: 995: 991: 988: 983: 982: 981: 980: 977: 973: 971: 964: 960: 954: 953: 952: 950: 947: 944:have exactly 943: 939: 927: 924: 916: 905: 898: 897: 896: 884: 881: 876: 875: 874: 873: 867: 863: 862: 861: 860: 857: 854: 841: 840: 837: 834: 830: 829: 828: 827: 824: 820: 818: 814: 810: 806: 802: 798: 794: 791:/3 such that 790: 782: 780: 779: 776: 769: 766: 762: 758: 754: 753: 752: 751: 748: 744: 736: 734: 731: 730: 727: 726:Andrew Kepert 719: 716: 713: 712:nowhere dense 709: 706: 703: 700: 697: 696: 695: 688: 683: 679: 678: 677: 670: 669:Michael Hardy 666: 665: 664: 662: 654: 649: 646: 642: 641: 640: 637: 635: 634:Michael Hardy 629: 627: 622: 620: 616: 605: 604: 601: 593:Image Problem 592: 590: 588: 577: 575: 573: 569: 566: 563: 557: 555: 553: 548: 541: 536: 532: 528: 524: 520: 516: 515: 514: 513: 509: 505: 500: 496: 493: 488: 487: 483: 481: 477: 473: 472:63.139.216.34 469: 461: 458: 457: 454: 439: 435: 431: 426: 425: 424: 423: 422: 421: 420: 419: 418: 417: 416: 405: 402: 398: 395: 394: 393: 392: 391: 390: 389: 388: 379: 376: 372: 371: 370: 369: 368: 367: 366: 365: 358: 355: 351: 350: 349: 348: 347: 346: 341: 338: 334: 333: 332: 331: 330: 329: 326: 320: 318: 317:nowhere dense 314: 308: 306: 302: 298: 291: 276: 267: 263: 262: 258: 254: 248: 245: 244: 241: 224: 220: 216: 215: 207: 196: 194: 191: 187: 186: 182: 176: 173: 170: 166: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 4346: 4343: 4340: 4337: 4334: 4331: 4327: 4323: 4286: 4267:47.44.96.195 4264: 4258: 4254: 4252: 4249: 4246: 4240: 4236: 4234: 4229: 4225: 4221: 4216: 4210: 4206: 4201: 4199: 4195: 4193: 4188: 4186: 4165: 4161: 4159: 3986: 3979: 3977: 3972: 3970: 3957: 3954: 3936: 3928: 3926: 3917: 3910: 3903: 3896: 3888: 3881: 3879: 3870: 3863: 3852: 3845: 3843: 3834: 3827: 3820: 3813: 3806: 3804: 3796: 3789: 3782: 3773: 3766: 3764: 3759: 3752: 3750: 3734: 3733: 3725: 3724: 3716: 3715: 3708: 3692: 3671: 3667: 3661: 3529: 3526: 3510: 3506: 3504:repetend. " 3501: 3499: 3495: 3492: 3487: 3484: 3480: 3476: 3473: 3447:— Preceding 3422:— Preceding 3419: 3400: 3381: 3377: 3357: 3353: 3332: 3326: 3318: 3316: 3311: 3310: 3299: 3294: 3292: 3281: 3268: 3266: 3261: 3259: 3240: 3237: 3232: 3230: 3225: 3224:The section 3223: 3202: 3200: 3195: 3191: 3189: 3186: 3183: 3178: 3176: 3171: 3170:The section 3169: 3131: 3127: 3123: 3119: 3114: 3110: 3105: 3101: 3096: 3092: 3087: 3083: 3035:— Preceding 3032: 2956: 2953: 2813: 2792:— Preceding 2788: 2784: 2781: 2612: 2343: 2203: 2145: 2087: 2067: 2041: 1991:JamesBWatson 1966: 1962: 1959: 1955: 1951: 1947: 1944: 1926: 1906: 1817: 1523: 1509: 1399: 1347:Yes. — Carl 1331: 1279: 1231:Uranographer 1229:interested. 1227: 1208: 1174: 1159: 1129: 1108:JWhiteheadcc 1098: 1072:JWhiteheadcc 1069: 1065: 1061: 1058: 1047:JWhiteheadcc 1043: 1027: 1013: 999: 974: 965: 961: 958: 948: 945: 941: 937: 931: 928:And so on... 922: 889: 865: 821: 816: 812: 808: 804: 800: 796: 792: 788: 786: 772: 757:measure zero 741:the page on 740: 732: 723: 692: 674: 658: 638: 630: 623: 611: 596: 585: 570: 567: 564: 561: 558:Nice linking 549: 545: 542:homeomorphic 523:Cantor space 501: 497: 491: 489: 485: 484: 466:— Preceding 462: 459: 450: 414: 396: 321: 309: 304: 295: 275:Chaos theory 252: 212: 148:Mid-priority 147: 107: 73:Mid‑priority 51:WikiProjects 34: 4253:Or perhaps 3958:Cantor dust 3762:is finite. 3666:Note: With 3384:Mike Rosoft 3205:explain it. 3126:to a value 2798:68.49.92.23 2050:Mike Rosoft 2017:—Preceding 1968:—Preceding 1889:Mike Rosoft 1849:Cheat notes 1843:—Preceding 1827:Cheat notes 1821:—Preceding 1512:81.134.83.6 1380:Mike Rosoft 1262:Mike Rosoft 1247:Mike Rosoft 1102:—Preceding 815:--and thus 747:Fresheneesz 375:Fresheneesz 337:Fresheneesz 313:perfect set 123:Mathematics 114:mathematics 70:Mathematics 4366:Categories 4215:Γ is also 3851:|, after | 3714:= {0, 1}; 3676:Nomen4Omen 3331:, it is a 1588:should be 1034:Azotlichid 1010:References 519:Cantor set 4164:and the 3934:| = 2-2. 3668:repeating 3664:repetend. 2641:. (Where 2512:power set 2322:whenever 1910:4.249.3.9 1654:SamIAmNot 1312:goatasaur 1298:goatasaur 1283:goatasaur 682:Gadykozma 661:Gadykozma 626:Gadykozma 619:Gadykozma 587:Gadykozma 39:is rated 4309:1234qwer 4306:1234qwer 3876:| + 2. 3449:unsigned 3424:unsigned 3416:Omission 3037:unsigned 2794:unsigned 2046:0.999... 2019:unsigned 1970:unsigned 1857:contribs 1845:unsigned 1835:contribs 1823:unsigned 1116:contribs 1104:unsigned 743:null set 737:Null set 714:- fixed. 468:unsigned 315:that is 3740:= ...; 3693:Is it? 2371:Dlivnat 2080:grmike 1376:perfect 987:Albmont 552:Zundark 301:page 53 255:on the 228:Systems 219:systems 175:Systems 150:on the 41:C-class 4170:Krauss 3938:Krauss 3887:| for 3805:where 3772:after 3403:345Kai 2366:is in. 2289:where 2070:Grmike 1372:closed 1016:Canter 976:CiaPan 775:Leocat 430:Zimri2 47:scale. 3869:| = | 3524:: --> 3523:: --> 3522:: --> 3488:: --> 3477:: --> 3295:seems 2914:and 2148:onto 2144:maps 1934:= 0.1 880:linas 853:Dzhim 833:linas 823:Dzhim 765:linas 600:Sasha 354:linas 325:linas 28:This 4354:talk 4271:talk 4174:talk 3975:. 3966:ref2 3962:ref1 3942:talk 3680:talk 3516:talk 3457:talk 3432:talk 3407:talk 3388:talk 3341:talk 3337:Daqu 3312:Also 3247:talk 3243:Daqu 3211:talk 3207:Daqu 3154:talk 3150:Bdmy 3140:talk 3136:Bdmy 3045:talk 2940:talk 2936:Daqu 2842:and 2802:talk 2768:talk 2764:Daqu 2375:talk 2074:talk 2054:talk 2048:. - 2027:talk 1995:talk 1978:talk 1914:talk 1893:talk 1876:talk 1853:talk 1831:talk 1803:talk 1658:talk 1516:talk 1384:talk 1378:. - 1374:and 1355:talk 1338:talk 1316:talk 1302:talk 1287:talk 1266:talk 1251:talk 1235:talk 1215:talk 1199:talk 1182:talk 1167:talk 1151:talk 1136:talk 1112:talk 1089:talk 644:not. 572:Deco 537:set" 531:talk 521:and 508:talk 504:Daqu 492:much 476:talk 434:talk 221:and 3964:or 3874:k-1 3826:= 3800:k-1 3788:= 3662:the 3530:All 3502:the 3324:" 3293:It 3203:not 3196:not 3064:lim 2966:lim 1872:CBM 1799:CBM 1351:CBM 1195:CBM 1085:CBM 946:one 923:end 817:not 305:The 247:Mid 142:Mid 4368:: 4356:) 4273:) 4232:" 4176:) 4157:. 4142:− 4097:− 4014:⋃ 3968:. 3944:) 3833:U 3795:U 3682:) 3672:un 3640:¯ 3637:02 3620:⋯ 3609:− 3601:⋅ 3587:− 3579:⋅ 3565:− 3557:⋅ 3518:) 3459:) 3434:) 3409:) 3390:) 3343:) 3279:" 3249:) 3235:" 3213:) 3181:" 3156:) 3142:) 3047:) 3014:∞ 2999:⋂ 2976:∞ 2973:→ 2942:) 2922:δ 2902:γ 2882:δ 2876:γ 2856:γ 2853:≤ 2850:δ 2830:δ 2827:≤ 2824:γ 2804:) 2770:) 2740:ℵ 2730:ℵ 2715:ℵ 2681:ℵ 2669:ψ 2649:ω 2629:ω 2626:≠ 2623:ψ 2599:ψ 2577:ψ 2573:ℵ 2544:ℵ 2466:ℵ 2418:ℵ 2397:ℵ 2377:) 2298:α 2266:… 2257:α 2247:α 2237:α 2227::= 2190:ℵ 2187:≥ 2171:⇒ 2112:ℵ 2076:) 2056:) 2029:) 1997:) 1980:) 1916:) 1895:) 1874:· 1859:) 1855:• 1833:• 1801:· 1780:≅ 1758:≅ 1694:∪ 1660:) 1652:. 1615:∪ 1518:) 1477:∪ 1449:≅ 1427:≅ 1386:) 1353:· 1340:) 1318:) 1304:) 1289:) 1268:) 1253:) 1237:) 1217:) 1197:· 1184:) 1169:) 1153:) 1138:) 1118:) 1114:• 1087:· 942:or 533:) 510:) 478:) 436:) 299:, 4352:( 4312:4 4269:( 4259:q 4255:Z 4241:q 4237:Z 4228:q 4222:q 4217:Z 4207:q 4202:Z 4198:q 4194:" 4172:( 4145:2 4137:1 4134:+ 4131:k 4127:2 4123:= 4118:k 4114:2 4110:+ 4106:| 4100:1 4094:k 4090:T 4085:| 4081:= 4077:| 4071:k 4067:T 4062:| 4039:m 4035:C 4029:k 4024:1 4021:= 4018:m 4010:= 4005:k 4001:T 3990:k 3987:C 3983:k 3980:T 3940:( 3932:k 3929:X 3921:8 3918:X 3914:4 3911:X 3907:3 3904:X 3900:2 3897:X 3895:| 3889:k 3885:k 3882:X 3871:X 3867:k 3864:X 3862:| 3856:1 3853:X 3849:k 3846:X 3838:1 3835:X 3831:2 3828:P 3824:2 3821:X 3814:k 3810:k 3807:P 3797:X 3793:k 3790:P 3786:k 3783:X 3777:1 3774:X 3770:k 3767:X 3760:k 3756:k 3753:X 3738:k 3735:X 3729:3 3726:X 3720:2 3717:X 3712:1 3709:X 3678:( 3646:3 3630:, 3626:0 3623:= 3617:+ 3612:6 3605:3 3598:2 3595:+ 3590:4 3583:3 3576:2 3573:+ 3568:2 3561:3 3554:2 3551:= 3548:4 3544:/ 3540:1 3514:( 3455:( 3430:( 3405:( 3386:( 3339:( 3329:q 3321:q 3317:" 3306:q 3302:q 3288:q 3284:q 3275:q 3271:q 3267:" 3245:( 3231:" 3209:( 3192:2 3177:" 3152:( 3138:( 3132:x 3130:( 3128:f 3124:x 3120:x 3118:( 3115:n 3111:f 3106:n 3102:A 3097:n 3093:f 3088:n 3084:A 3068:n 3043:( 3019:. 3009:1 3006:= 3003:m 2970:m 2938:( 2879:= 2800:( 2766:( 2762:. 2744:0 2735:2 2726:= 2719:0 2710:2 2685:0 2676:2 2672:= 2613:ψ 2548:0 2539:2 2535:= 2532:c 2496:P 2470:0 2461:2 2457:= 2453:| 2449:C 2445:| 2422:0 2373:( 2354:x 2344:C 2330:x 2310:1 2307:= 2302:n 2274:2 2270:) 2261:3 2251:2 2241:1 2233:. 2230:( 2224:) 2221:x 2218:( 2215:f 2204:C 2183:| 2179:C 2175:| 2168:] 2165:1 2162:, 2159:0 2156:[ 2146:C 2132:f 2109:= 2105:| 2101:C 2097:| 2072:( 2052:( 2025:( 1993:( 1976:( 1940:3 1936:3 1932:3 1930:/ 1912:( 1891:( 1878:) 1870:( 1851:( 1829:( 1805:) 1797:( 1783:C 1777:) 1774:C 1771:( 1766:R 1762:f 1755:) 1752:C 1749:( 1744:L 1740:f 1719:C 1716:= 1713:) 1710:C 1707:( 1702:R 1698:f 1691:) 1688:C 1685:( 1680:L 1676:f 1656:( 1640:C 1637:= 1634:) 1631:C 1628:( 1623:R 1619:f 1612:) 1609:C 1606:( 1601:L 1597:f 1576:C 1573:= 1570:) 1567:C 1564:( 1559:R 1555:f 1551:= 1548:) 1545:C 1542:( 1537:L 1533:f 1514:( 1496:) 1493:C 1490:( 1485:R 1481:f 1474:) 1471:C 1468:( 1463:L 1459:f 1455:= 1452:C 1446:) 1443:C 1440:( 1435:R 1431:f 1424:) 1421:C 1418:( 1413:L 1409:f 1382:( 1357:) 1349:( 1336:( 1314:( 1300:( 1285:( 1264:( 1249:( 1233:( 1213:( 1201:) 1193:( 1180:( 1165:( 1149:( 1134:( 1110:( 1091:) 1083:( 1028:" 949:1 938:1 934:∞ 919:2 914:. 912:1 908:1 901:0 893:n 849:3 844:3 813:Q 809:Q 805:n 801:k 797:k 793:n 789:k 529:( 506:( 474:( 432:( 277:. 259:. 225:. 154:. 53::

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