Set Theory and Circumference of a Circle

[Saphiraflames]

Hey friends!

How can the set of all points on the circumference of a circle be an Infinite Set?

Anyone could explain?

Thanks in advance!

-Saphira :)

 

[Hurkyl]

Hey friends!

How can the set of all points on the circumference of a circle be an Infinite Set?

Anyone could explain?

Thanks in advance!

-Saphira :)

Erm, because you can count them. The number is c -- the same as the cardinality of the set of all real numbers. (or, more suggestively, the same as the cardinality of the set $[0, 2\pi)$) And even if you don't count them, it's not hard to show that for any finite set of points on the circle there exists another point not in that set.

 

[Saphiraflames]

Erm, because you can count them. The number is c -- the same as the cardinality of the set of all real numbers. (or, more suggestively, the same as the cardinality of the set $[0, 2\pi)$) And even if you don't count them, it's not hard to show that for any finite set of points on the circle there exists another point not in that set.

Hurkyl, what is cardinality in simple terms? And can the number of points be counted by computers?

 

[yossell]

At the core of the arithmetic of cardinalities is the idea that two sets are the same size if and only if there is a 1-1 correspondence between the two the sets.

For instance, in the finite case, given some Fs and given some Gs, you know there are as many Fs and Gs if you can lay each F against one and only one G and, at the end of this process, every F is associated with some G and every G has been associated with some F. However, there are more Fs than Gs if, no matter how you pair an F with a G, you always end up with some Fs left over.

Clearly, this notion of pairing gets the right results when we are dealing with finite sets. The key ideas in the infinite case is to extend this very definition of sameness of size to infinite sets. In general: F and G are the same size - more technically: have the same cardinality - if there is a 1-1 function from all the Fs to all the Gs.

This has some counterintuitive consequences in the infinite case: the even numbers can be put in 1-1 correspondence with all the integers:
1 2 3 4 5 6...
2 4 6 8 10 12...

Each integer in the top corresponds to one and exactly one even below, and every even number is associated some integer by this pairing.

It turns out that the rational numbers, numbers of the form m/n, can also be put into 1-1 correspondence with the integers!

At that point, you might have thought there was nothing interesting in the notion - a infinite set was just infinite, and all infinite sets, being infinite, could all be put in 1-1 correspondence to each other. But no! The point at which the theory gets mathematically interesting is that NOT all infinite sets can be put in a 1-1 correspondence; indeed, the cardinality of the natural numbers is the smallest size of infinity; moreover, the reals are just such a set for which it is provably impossible to map 1-1 into the integers. Since there are clearly mappings from a subset of the reals to the natural numbers, we say that such a set is strictly larger than the size of the natural numbers.

It turns out that the infinite cardinalities, under the right set-theoretic assumptions, form a well ordering: after the size of the natural numbers, often called aleph-zero, there's a successor cardinality which is called aleph-one. And so on.

Given an infinite set, it's always possible to form a bigger set, just by taking the set of all subsets of the original set. If the original cardinality is k, the cardinality of the set of all subsets of k is often called 2^k. The cardinality of the real numbers is the same as the cardinality as all the subsets of the natural numbers, so is often written 2^{aleph-zero}.

Is 2^{aleph-zero} the next size up? Is it aleph-one? Or is it bigger?

After a lot of hunting, this turned out to be (provably) unprovable in the set-theories that we have any kind of confidence in. Some feel that if something so basic about infinities cannot be answered, we don't have such a great mathematical grip on the notion of infinity. Others accept that, even in mathematics, there can be uncertainty.

Whichever way you go, I wouldn't say that these infinite sets are the kinds of things that could be counted or calculated by computer.

 

[vici10]

As Hurkyl said, cardinality of the set of all points of the circle is equal to cardinality of $$[0,2\pi)$$. This means that there is one-to-one correspondence from $$[0,2\pi)$$ onto circle that can be represented as $$\{(x,y) : x=R\cos \phi , y=R\sin \phi, 0 \leq \phi < 2\pi\}$$. The corespondence is $$(x,y) \rightarrow \phi$$.

Now, there is one to one correspondence from $$[0,2\pi)$$ onto set of all non-negative real numbers $$\{ r \in R : r >= 0\}$$. The corresndence is $$\phi \rightarrow tan(\phi/4)$$.

You know that the set of all non-negative real numbers is infinite hence the set of all points of the circle is infinite.

 

[HallsofIvy]

Erm, because you can count them.

No, you can't count them- they are uncountable!

The number is c -- the same as the cardinality of the set of all real numbers. (or, more suggestively, the same as the cardinality of the set $[0, 2\pi)$) And even if you don't count them, it's not hard to show that for any finite set of points on the circle there exists another point not in that set.

 

[SW VandeCarr]

The OP specified a point set. If it's a point set, the cardinality should be $$\aleph_0$$. If it's a continuum, the cardinality should be C. No?

 

[CRGreathouse]

The OP specified a point set. If it's a point set, the cardinality should be $$\aleph_0$$. If it's a continuum, the cardinality should be C. No?

Why would you be able to say that an arbitrary set of points would be countable?

 

[SW VandeCarr]

Why would you be able to say that an arbitrary set of points would be countable?

A point is discrete and I'm thinking they could be placed in a one to one correspondence with the integers just as the rational numbers can. In other words, for any point between two points there exists a rational number to which it corresponds.

EDIT: If a point set has the cardinality C, why even distinguish a point set from the continuum?

 

[yossell]

The OP specified a point set. If it's a point set, the cardinality should be $$\aleph_0$$. If it's a continuum, the cardinality should be C. No?

The OP specified a set of points. There's no reason to think a set of points must be countable. Indeed, as has been shown, the set of points on the circumference of a circle is uncountable.

 

[SW VandeCarr]

The OP specified a set of points. There's no reason to think a set of points must be countable. Indeed, as has been shown, the set of points on the circumference of a circle is uncountable.

I agree there's no reason to assume they must be countable. I was trying to account for Hurkyl's question, "..because they are countable?" in the light of the OPs question. I suppose a point set could be countable such as in a discrete topology or as a one dimensional set of all lattice points, corresponding to all rational numbers in $$[0,2\pi)$$ on the circle, that is dense in the reals.

EDIT: That of course would not be the set of all points on the circle.

 

[CRGreathouse]

EDIT: If a point set has the cardinality C, why even distinguish a point set from the continuum?

Indeed, that was precisely my question of you.

 

[SW VandeCarr]

Indeed, that was precisely my question of you.

See my previous post 11. Thanks.

 

[GoldPheonix]

Definition 1.) Cardinality. Loosely speaking, it refers to the "size" of a set. This is to say, cardinality is the number of objects (often technically labeled "elements") contained in a set. So if a set, X, is {a,b,c,d}, then the cardinality of X is 4, because it has four objects contained in it; likewise, the cardinality of our alphabet is 24, because we have 24 distinct letters in our language.

Theorem 2.) Two sets of objects (let's call them A and B) are said to have the same cardinality, if and only if, they have "1-to-1 correspondence." This is to say, if every object in A can be attached to a unique object in B, and vice-versa. An intuitive example would be: "Every knight has a horse, and every horse has a knight." Let the first set, A, represent some set of horses and let the second set, B, be represented by a set of knights. Even if we haven't counted the set of horses or the set of knights, we can know that they have an equal number of horses and knights if we can't mount every knight to a horse and have no horses or knights left over.

Definition 3.) The cardinality of a set is said to be (uncountably) infinite, if you can find a "1-to-1 correspondence" between every point in the set and an interval of the real line (I'm being slightly incorrect here, but it wouldn't be beneficial to go into why).

This is to say, the cardinality of [2,3] (the set of numbers greater than 2 but less than 3) is infinite (So, for an intuitive proof: I can find you some other numbers that are only slightly larger than 2, mainly: 2.1, 2.01, 2.001, ad infinitum that fits between 2 and 3, and since there are an infinite number of those, there must be, as a collection of them, an infinite cardinality).Proposition 4.) I can uniquely identify every point on the circumference of a circle with an angle (ranging between 0 and 360 degrees, if you like; or 0 and 2 pi, if you know about radians), and visa versa. This should be intuitively true, because the angle, going out from the center, points to a unique point on the circumference of the circle. The angle, however, is an interval of real numbers [0,360).

Conclusion.) Taking our previous theorem, our definition of what cardinality means, and our definition of what it means to have an (uncountably) infinite cardinality, it follows logically that since I can pair every point uniquely to an angle, and the set of all angles is (uncountably) infinite --that therefore, the number of points on a circumference is infinite.
(This is a more intuitive discussion, I think, of the principles that you're asking about)