20 replies to this topic

[bonanova]

Consider the unit cube x, y, z = [0,1].
Does it contain more points than its edge x = [0,1], y=z=0?

Edit. Let's state it simply:
Does a cube contain more points than are contained in one of its edges?

[Rob_Gandy]

Spoiler for I would say

yes. Even though both are infinite I would say the number of points in the cube is larger. Similar to how I would say the number of integers is larger than the number of natural numbers.

[mmiguel]

Consider the unit cube x, y, z = [0,1].
Does it contain more points than its edge x = [0,1], y=z=0?

Edit. Let's state it simply:
Does a cube contain more points than are contained in one of its edges?

Spoiler for

yes,
though both are uncountably infinite sets of points, an edge of a cube is a subset of the cube itself, so the cardinality of the edge is less than or equal to the cardinality of the cube. it is not equal, because there exist points which belong to the cube, but do not belong to the edge. therefore, the only remaining option is that the cardinality of the edge is less than the cardinality of the cube.

[EventHorizon]

Spoiler for My answer

First lets look at the degree of infinity of the number of points in each. The number of points in the edge has the same degree of infinity as the same as the number of real numbers, which is 2^Aleph₀. If you look at where that 2 came from, it really is just any finite number greater than 1. You can see this easily by noting that n*Aleph₀=Aleph₀, where n is any positive finite number. If you assume the number of points on an edge was x, then the number of points in the cube would be x³. Substituting 2^Aleph₀ for x gives, (2^Aleph₀)³ = 2^3*Aleph₀ = 2^Aleph₀. So the degree of infinity of the number of points in the edge are the same as that of the number of points in the cube.

The cube contains 12 edges, so it certainly contains more points... right? That's even ignoring the internal points and points on the faces of the cube! The degree of infinity is the same, but there are surely points more in the cube because it contains the edge and more. (infinite copies of the edge even! You can let y and z be any constant value in [0,1] to get another copy)

So the answer is yes if only because every point in the edge is in the cube and not vice versa. (e.g., a set contains more elements than any strict subset of itself)

[bonanova]

mmiguel, would you say there are different cardinal numbers for one edge, 12 edges, faces, solids?

BTW just to clarify, I mean by cube to denote a solid, as per EH's reasoning.

Any other views?

[bushindo]

My take on this

Spoiler for

There are the 'same' number of points on the cube as there are on the edge. That is because we can construct a 1-to-1 mapping between any point on the cube and the corresponding point on the edge.

Let (x,y,z) be the 3-dimensional coordinate of any point in the cube (surface included) where x, y, z = [0,1]. In decimal form, let's write x as

x = x₀.x₁x₂x₃x₄x₅... where x₀ = 0 or 1 is the number before the decimal point, and x_i = 0, 1, ..., or 9 for i > 0. (Of course, if x₀ = 1, then x_i must all be equal to 0)

Similarly, we can write y and z in the same decimal form,

y = y₀.y₁y₂y₃y₄y₅...
z = z₀.z₁z₂z₃z₄z₅...

Now, we can establish a 1-to-1 relationship between any point (x,y,z) on the cube to a point e on line segment [0, .111 ]. To do that, we can simply interleave digits and construct e from (x,y,z) as follows

e = .x₀y₀z₀x₁y₁z₁x₂y₂z₂...

It is easy to see that we can reverse the process and recover a 3-D coordinate in the cube (x,y,z) given a point e. Now that we have a strict 1-to-1 matching between points on a cube to the segment [0, .111], we can do a strict 1-to-1 matching between any point e in [0, .111] and any point s in [0, 1] using a linear function

s = e * (1/.111 )

[vinay.singh84]

Spoiler for My answer

To solve this, one needs cardinal numbers.
The edge is an uncountable set of points with specific cardinality of Aleph₁.
This can be shown by creating a simple bijection y=tan(pi*(x-0.5)) of (0,1) to R.
Similarly, each plan contains an uncountable number of edges and the cube and an uncountable number of planes.
Thus, the cardinality of the set of points in the cube == Aleph₁*Aleph₁*Aleph₁ == Aleph₁
So the cube contains the same number of points as any of its edges.

[bonanova]

And has the title become germane?

[mmiguel]

mmiguel, would you say there are different cardinal numbers for one edge, 12 edges, faces, solids?

BTW just to clarify, I mean by cube to denote a solid, as per EH's reasoning.

Any other views?

Spoiler for

i was interpreting the OP to mean solid.
i did a little research, and it seems that by the standard mathematical definition of cardinality, it seems that just because an infinite set A is a superset of an infinite set B, this does not mean that |A| > |B|.

the concept that the superset operator translates into a "greater-than" inequality when going from sets to the cardinalities of those sets is true for finite sets, as is the bijective mapping view of cardinality that mathematicians have chosen to stick to when generalizing from finite sets to infinite sets.

under this generalization, the two concepts are no longer equivalent.
this is something i may have read a long time ago, but did not recall when providing my answer.

http://en.wikipedia....iki/Cardinality
"Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite setS that have the same size as S, although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it."

so what this means is that a solid cube has the same cardinality as an edge of the cube because a bijective mapping can be formed between the two sets.

i wonder, is there a standard definition for the infinite-set generalization of the subset/superset interpretation of cardinality from the finite case?
perhaps such a definition was deemed to be not useful...

thinking over this, it is all so very strange:
the cube contains all the points that the edge contains, and then some....
yet each point in the cube can be paired with a point in the edge, with no points in the edge being left unpaired...

if these are both true, than this must be a contradiction (mustn't it?).
if one is false, then why is it false?

i think it must be a contradiction if both are true because suppose we set about pairing points in the cube to the edge.
let's say we restrict ourselves to first pair all the points in the edge of the cube before pairing any other points of the cube.
since the cube contains the edge, we could pair each point of the edge from the cube set perspective, to the same point in the edge set perspective.
each point in the edge gets paired with itself. thus there are no longer any points in the edge that are unpaired. but there are points not on the edge, which still are part of the cube which are not paired to any points within the edge, and cannot be, since all the points in the edge set are already paired (with themselves from the cube set perspective). hence no bijective mapping can be formed if restrict ourselves to pairing the points within the edge first in the cube set perspective....

perhaps someone can enlighten me?

Edited by mmiguel, 18 October 2012 - 03:34 AM.

[mmiguel]

sorry, thought a little more, wanted to write a little more

Spoiler for

is being able to form a 1-1 mapping really equivalent to having the same number of things?
obviously yes for the finite case

in the infinite case, it seems to me that this definition of cardinality is more like: starting from the cube set or the edge set, there are always enough available point resources in the other set to support a mapping no matter how many points you ask for a partner for (i.e. there exists an injective mapping).

this doesn't to me seem to mean that when you have paired all the points in your starting set, that there are no longer any point resources in the other set left (and are therefore unpartnered) i.e. does not imply that the mapping is surjective.

to prove surjectivity, for example in bushindo's case, you invert the algorithm and show that it is injective when starting from the other set as well.

the crux of the counter-intuitiveness is that one real number, can contain the information of three real numbers, as bushindo showed.
what if there is an infinite amount of information? (by information i mean the ordered set of decimal digits making up a number) ----> what if x=y=z=pi or some other transcendental number?

i guess e would be another transcendental number, but everything would still work.

but this almost seems like it's cheating. logically what we are doing is mapping one number to three numbers using a clever encoding scheme.

any real number can be thought of as an infinite ordered set of natural numbers (the digits of the number).
by saying we can construct the number e, from numbers x,y, and z, we are implicitly assuming that e has the capacity to store x, y, and z.
this assumption is logically equivalent to the conclusion that a single edge has the capacity to provide partner points to every point in a cube.
doesn't that make assuming that e can be constructed from x,y, and z, to prove that edge and cube cardinalities are the same, a circular fallacy?
if we are not allowed to assume that e has the capacity to completely store x, y, and z, but rather the capacity to store only one of them, then I think we would conclude the cardinality of the cube is greater than the edge.

so how do we prove that e can be constructed for any real x,y,z?
perhaps the argument is, I can try to construct it and I know what I would do at every step, so therefore it is constructable?
but if x,y, or z is transcendental, then such a construction would never end....
well at least it couldn't be described in terms of a simple repetitive instruction loop (e.g. the rest of the digits are all zero, or the rest of the digits are 142857 repeating). i suppose it's always an infinite construction even in the simplest of cases.

perhaps the concept that a single real number can store the information of an arbitrary number of other real numbers is an axiom rather than something proven from simpler axioms. if that is the case, then the original question is essentially asking whether we choose to adopt this axiom or not.

anyone care to offer some help?

Edited by mmiguel, 18 October 2012 - 04:15 AM.