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"Fat" Cantor Sets: do these paradoxical objects bother anyon

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How can 1/2 be equal to 1? For the uninitiated, here is a brief construction of a so-called "Fat Cantor Set":

-Start with the closed interval [0,1]; intuitively speaking, begin with a foot-long ruler.

-Remove the middle 1/4 from this interval. (More precisely, remove from here an open interval of length 1/4 centered at the middle of [0,1].)

-From the two remaining intervals ( [0,3/8] and [5/8,1] ), take away 1/16 from the middle of each. That is, take from each an *absolute* 1/16, NOT 1/16 of what remains. (This distinction is important, as it will affect the result: for example, from [0, 3/8] what you DO want to do is take away an open interval of length 1/16 = .0625; what you do NOT want to do is take away an open interval of length (1/16)*(3/8) = .09375.)

-Continuing this process inductively will yield, at step n, 2^n sub-intervals, and from each of these you will take away a piece of length (1/4)^(n+1).

The result of this process is that the sum of the lengths of the pieces removed will be 1/2. However, the union of these removed intervals will be dense in [0,1].

In other words, if you look at the pieces removed, all together they "look like" the original foot-long ruler, but they only sum up to one-half this length.

Edited by Epistemological Engineer
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How can 1/2 be equal to 1? For the uninitiated, here is a brief construction of a so-called "Fat Cantor Set" [...] In other words, if you look at the pieces removed, all together they "look like" the original foot-long ruler, but they only sum up to one-half this length.

This doesn't bother me very much, but that may be because I've done a lot with analysis, and so I've incorporated paradoxical sets and functions like this into my thinking.

Also, a simpler example of a set containing no intervals but with positive measure is: take the unit interval [0, 1] and remove the rational numbers. The resulting set (the irrational numbers between 0 and 1) has measure 1, even though the set contains no interval (since every open interval contains a rational number).

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Also, a simpler example of a set containing no intervals but with positive measure is: take the unit interval [0, 1] and remove the rational numbers. The resulting set (the irrational numbers between 0 and 1) has measure 1, even though the set contains no interval (since every open interval contains a rational number).

No no no, the part that bothers me is not the Fat Cantor set itself, but its complement. See, the rationals in [0,1] cannot be written as a union of disjoint open intervals with positive measure; the complement of the FC set can.

"Contradictions cannot exist in nature"--this must mean that nature is discrete, no? Of course, a paradox is not necessarily a contradiction (?), but it at least appears to be contradictory, which disturbs me nonetheless.

Edited by Epistemological Engineer
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Well first, by no theory does 1/2 = 1 but rather, the set of rationals is equinumerous with the set of positive integers. Second, this is no stranger than the proof that the set of rationals between 0 and 1 is equinumerous with the set of all rationals, nor stranger than the proof that the set of reals between 0 and 1 is equinumerous with the set of all reals.

Think of it this way: The set {even numbers} is enumerable and the set {odd numbers} is enumerable. And the union of the sets, that is to say, the positive integers are enumerable. But the set {{evens}, {odds}} is not enumerable. It is of magnitude aleph_1.

Third, I have very serious doubts as to whether even the rational numbers are genuine numbers--for one thing, they don't count anything. To me, counting is an essentially numberish property. But this point could quickly devolve into semantics. The real problem is that I have serious doubts about whether the real numbers actually measure anything in the real world. As far as I can tell, all that we can say with certainty about them is that they fill certain equations such that they provide useful though not necessarily accurate answers. For instance, a perfect circle could not truly exist with our common understanding of physical reality (if there is some kind of quantum reality that I don't know about, then maybe this line of thinking is false). Yet we often treat figures as if their borders were composed of transfinite set of points equidistant from a central point--and doing so makes equations quick and easy, though not in perfect alignment with physical reality.

So none of this bothers me at all. What partially bothers me is the objectivity of numbers.

[Editted for grammar.]

Edited by aleph_0
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No no no, the part that bothers me is not the Fat Cantor set itself, but its complement. See, the rationals in [0,1] cannot be written as a union of disjoint open intervals with positive measure; the complement of the FC set can.

So the paradox is the fact that there exists a set which is dense in [0, 1], but at the same time doesn't have measure 1? I guess I don't understand why the compliment of the Cantor set you've constructed is so troubling while that Cantor set itself isn't.

Regardless, if it's the compliment you're worried about for whatever reason, then you're right that my example doesn't do the job, since it's not open.

"Contradictions cannot exist in nature"--this must mean that nature is discrete, no? Of course, a paradox is not necessarily a contradiction (?), but it at least appears to be contradictory, which disturbs me nonetheless.

I'm not sure what you mean by "nature must be discrete." In any case, since mathematics deals with concepts of method, the ideas in mathematics formed to give meaningful measurements of reality need not themselves correspond to reality, if that's what you're worried about (e.g., any kind of infinity in mathematics, or the imaginary unit).

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Third, I have very serious doubts as to whether even the rational numbers are genuine numbers--for one thing, they don't count anything.

Rational numbers count units which are regarded as equal parts of a whole unit. "7/30" means you've cut something up (whether mentally by selective focus or physically with a knife) into 30 pieces, and you've counted out 7 of them. Or, if you like, 0.4 inches means you've subdivided an inch into a new unit (dividing the inch into ten equal parts) and have measured out 4 of them

Also, what issue do you take with the objectivity of numbers?

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Think of it this way: The set {even numbers} is enumerable and the set {odd numbers} is enumerable. And the union of the sets, that is to say, the positive integers are enumerable. But the set {{evens}, {odds}} is not enumerable. It is of magnitude aleph_1.

You miss the ball here: the set {{evens},{odds}} is enumerable, it has cardinal number aleph_0 and has Omega+Omega as ordinal number. For the set {{evens},{odds}} stands in 1-1 correspondence with {natural numbers}.

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Rational numbers count units which are regarded as equal parts of a whole unit. "7/30" means you've cut something up (whether mentally by selective focus or physically with a knife) into 30 pieces, and you've counted out 7 of them. Or, if you like, 0.4 inches means you've subdivided an inch into a new unit (dividing the inch into ten equal parts) and have measured out 4 of them

Also, what issue do you take with the objectivity of numbers?

What of objects which do not have 30 equal parts? For instance, a most basic unit of matter.

That numbers are objects seems highly dubious to me, though somewhat difficult to refute.

You miss the ball here: the set {{evens},{odds}} is enumerable, it has cardinal number aleph_0 and has Omega+Omega as ordinal number. For the set {{evens},{odds}} stands in 1-1 correspondence with {natural numbers}.

You're right, what I meant is that the set {2, 4, 6, 8... 1, 3, 5, 7...} or {{evens} followed by {odds}} is not enumerable. Damn my computability substitute professor for this misinformation.

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So the paradox is the fact that there exists a set which is dense in [0, 1], but at the same time doesn't have measure 1?

No, it's more than just this: not only is (FC complement) dense in [0,1] with measure less than 1, it is also constructed from open intervals with positive measure. The set of rationals in [0,1] does not have this latter property.

Not to be rude to everyone else, but I think that the discussion of the countability and ordinals here is not quite to the point. The paradox I'm trying to examine is the idea that you can take a ruler, remove from it pieces that have nonzero length all of which add up to 1/2, and have no intervals left over.

Actually, Nate, the FC set itself bothers me as well: it is nowhere-dense, yet has positive measure. Maybe I could spare some excess verbiage by focusing on the "dust" itself ;-)

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Actually, Nate, the FC set itself bothers me as well: it is nowhere-dense, yet has positive measure. Maybe I could spare some excess verbiage by focusing on the "dust" itself ;-)

Well, another way to think about it is that one could never actually carry out such a construction in reality, since it involves (countably) infinite unions and intersections.

I just look at counterintuitive things like this as the price you have to pay for a general-enough theory that's able to handle countable unions and intersections.

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What of objects which do not have 30 equal parts? For instance, a most basic unit of matter.

Well, one wouldn't use fractions to measure those. But there are plenty of things for which it would be impractical to reduce to quanta (like pizzas and rulers, among others). And rational numbers work quite well for measuring those things. Just because some objects cannot be split up into any fraction one desires does not mean that fractions have no purpose.

That numbers are objects seems highly dubious to me, though somewhat difficult to refute.

Well, if by "object" you mean a material existent, you're right (except for the part about it being difficult to refute). That doesn't make numbers unreal or non-objective, though, since they are concepts, which are objective (if formed correctly).

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Well, another way to think about it is that one could never actually carry out such a construction in reality, since it involves (countably) infinite unions and intersections.

This may be the only way out. (That is, my only way out of going out of my mind over this :-) )

I'll think about the stuff you've said, and perhaps add more in the future. The not-countably-infinite future, that is.

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