Infiniter?

[PrinceofLowLight]

Alright, there's an infinite number of rational numbers. But there's an infinite number of irrational numbers between every two rational numbers.

They're both infinite sets. But isn't the set of irrational numbers kind of...inifiniter?

[Enola Straight]

Some infinities are biger than others.

Lets take a number line from 0 to 10.

There are an infinite number of points along a line segment.

An infinite number of points between o and one, also an infinite number between 0 and ten.

Thus, the second infinity is ten times larger than the first.

This is, IIRC, how renormalization works in Quantum Mechanics.

[Ariphaos]

An infinite number of points between o and one, also an infinite number between 0 and ten.

Thus, the second infinity is ten times larger than the first.

Incorrect.

Every number you can find from 0 to 10 can uniquely map to a real number between 0 and 1.

Aleph-naught infinity is the 'smallest infinity'. The set of all rational numbers is an example of aleph-naught infinity, and it is exactly the same scale as the set of positive integers, because for every rational number, you can order it such that you assign each a unique positive integer, even though there are clearly more rational numbers than positive integers (infinitely more, even).

You cannot do this with real numbers, thus, they are considered to have aleph one infinity. Likewise, the set of all sets cannot be mapped to real numbers, so that construct is considered aleph two. AFIAK there is no known construct for an aleph three infinity.

[Surlethe]

Alright, there's an infinite number of rational numbers. But there's an infinite number of irrational numbers between every two rational numbers.

They're both infinite sets. But isn't the set of irrational numbers kind of...inifiniter?

Yes. The set of rational numbers is countable, which means you can set up a bijection -- a one-to-one and onto function -- between it and the whole numbers {1, 2, 3, 4, 5, 6, 7, 8, ...}. The method is rather clever, actually:

Code: Select all

         1    2    3    4    5    6   7    8    9    10  ...
        -------------------------------------------------------
     1|1/1  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10 ...
     2|2/1  2/2  2/3  2/4  2/5  2/6  2/7  2/8  2/9  2/10 ...
     3|3/1  3/2  3/3  3/4  3/5  3/6  3/7  3/8  3/9  3/10 ... 
     4|4/1  4/2  4/3  4/4  4/5  4/6  4/7  4/8  4/9  4/10 ...
     5|5/1  5/2  5/3  5/4  5/5  5/6  5/7  5/8  5/9  5/10 ...
     6|6/1  6/2  6/3  6/4  6/5  6/6  6/7  6/8  6/9  6/10 ...
     7|7/1  7/2  /73  7/4  7/5  7/6  7/7  7/8  7/9  7/10 ...
     8|8/1  8/2  8/3  8/4  8/5  8/6  8/7  8/8  8/9  8/10 ...
     .   .    .    .    .    .    .    .    .    .    . 
     .   .    .    .    .    .    .    .    .    .    . 
     .   .    .    .    .    .    .    .    .    .    . 

So the key to setting up a bijection between the whole numbers and the rationals is to find a way to list them in order, to say "this is the first fraction; this is the second; etc". To list the fractions, start in the upper-left-hand corner of the square, and follow a zig-zag through the graph, eliminating fractions which have already been listed (to ensure bijectivity). You'll end up listing the fractions like so: {1/1, 2/1, 1/2, 3/1, 1/3, 4/1, 3/2, 2/3, 1/4, ...} Since you can list the fractions, there exists a bijection between Q and N.

Now, let's consider the cardinality of the reals (R). We'll take for granted the facts

1. Every real can be represented by a decimal expansion; and
2. a set with an uncountable subset is uncountable.

So if we show the reals in the interval [0,1] are uncountable, then the reals as an entirety are also uncountable (if an infinite set is uncountable then there is not a bijection between it and N). So let's assume for contradiction the reals on [0,1] are countable, which means we can arrange them in a list. Using decimal expansions,

0.0000000000000000000000...
0.a_11a_12a_13a_14a_15...
0.a_21a_22a_23a_24a_25...
0.a_31a_32a_33a_34a_35...
.
.
.
0.a_n1a_n2a_n3a_n4a_n5...
1.000000000000000000000...

where the ith real r_1 in [0,1] is expressed by the decimal expansion r_i = 0.a_i1a_i2a_i3a_i4... .

Now, keeping in mind we're assuming we've just listed all of the reals between 0 and 1, we're going to show there's a real we haven't listed. Let's make an algorithm: change a_ii (i.e., the ith digit of the ith number) to 4 if a_ii ≠ 4, and change it to if a_ii = 4. Then we have created a decimal between 0 and 1 which was not listed above, because it is different from every other real listed in at least one digit (the jth digit ≠ the jth digit of the jth real because we changed it!). But we assumed we listed all of the reals between 0 and 1; thus, we have reached a contradiction! Hence, the set of reals between 0 and 1 is uncountable; and so |R| > |Q|.

[Kuroneko]

Alright, there's an infinite number of rational numbers. But there's an infinite number of irrational numbers between every two rational numbers.

Yes, but also vice versa--there is an infintie number of rational numbers between every two irrational numbers as well. Both the rationals and irrationals are "dense" in the reals. That fact alone doesn't mean much. The cardinality of a set is its "size" in the sense of "how many" elements it contains. Explicitly, sets A and B have the same cardinality iff there exists a one-to-one correspondence between A and B. For example, suppose Q is the set of rationals and N is the set of positive integers. One way to construct a correspondence is as follows. Any rational r can be uniquely represented as (-1)^k[n/m] with k = 0 or 1, n≥0, m>0 integers with n,m sharing no divisors other than 1, so that for each rational r, there is a unique number [2^k][3^n][5^m]. Sort these numbers in increasing order: (1) 10,15,25,30,40,45,50,..., corresponding to the rationals (2) 0/1,1/1,1/2,-1/1,-1/2,2/3,-1/2,... (e.g., 45 = [2^0][3^2][5^1], corresponding to (-1)^0*2/3 = 2/3). Therefore, rationals and positive integers have the same cardinality, since for any position in list (1) there is a unique rational, and vice versa.

They're both infinite sets. But isn't the set of irrational numbers kind of...inifiniter?

Yes, but not for the reason you state. A classic diagonal cut can be used to prove this fact. Suppose there is a one-to-one correspondence f(n) between the naturals and the reals in [0,1). Represent the reals in [0,1) with decimal (or binary or another base, if you prefer) digits with ending 9's disallowed [1], so that
f(1) = 0. a11 a12 a13 a14 ...
f(2) = 0. a21 a22 a23 a24 ...
f(3) = 0. a31 a32 a33 a32 ...
with every real in [0,1) eventually showing up in this list represented as decimal digits. Construct the number B = 0. b1 b2 b3 b4 ..., with digit b_n defined as 0 if ann [the nth digit of f(n)] is nonzero and 1 if ann is 0. The trick is that this number B is different from every f(n), as it evident from the fact that for any n it differs from f(n) by at least one digit (the nth in particular). Our assumption of there being a correspondence must be faulty, so there can be no such correspondence. Hence, there are "more" reals in [0,1) [and consequently more reals] than natural numbers.

[1] This is done so that the representation will be unique. For example, 0.20000... = 0.199999....; the latter representation will be disallowed. In base n, ending in (n-1)'s would be disallowed instead.

Some infinities are biger than others. Lets take a number line from 0 to 10. There are an infinite number of points along a line segment. An infinite number of points between o and one, also an infinite number between 0 and ten. Thus, the second infinity is ten times larger than the first.

No. Its Lebesgue measure is ten times larger than the first, but they have the exact same cardinality. This is obvious because f(x) = x/10 maps [0,10] to [0,1] and is a one-to-one correspondence.

This is, IIRC, how renormalization works in Quantum Mechanics.

No.

Aleph-naught infinity is the 'smallest infinity'. The set of all rational numbers is an example of aleph-naught infinity, and it is exactly the same scale as the set of positive integers, because for every rational number, you can order it such that you assign each a unique positive integer, even though there are clearly more rational numbers than positive integers (infinitely more, even).

Yes, although the sense that there are "clearly more" is rather suspect.

You cannot do this with real numbers, thus, they are considered to have aleph one infinity. Likewise, the set of all sets cannot be mapped to real numbers, so that construct is considered aleph two. AFIAK there is no known construct for an aleph three infinity.

Ah, no. First, there is no such thing as a set of all sets in any extension of ZF (the standard set theory), ZFC included (one needs the axiom of choice to order the cardinalities as alephs). There's a "class" of all sets in set theories that can talk about such things (e.g., vNBG set theory), but it does not have a cardinality at all. If one takes the generalized continuum hypothesis as an axiom, then an example of an aleph-3 set is P³(N) = P(P(P(N))), where P() is the power set of and Z is the set of natural numbers. Note that even without the generalized continuum hypothesis |P³(N)|≥aleph-3.

[Surlethe]

More generally, the set X and its power set P(X) do not have the same cardinality. Suppose, for contradiction, there is a bijection f:X->P(X) between X and P(X). Now, given an element x in X, x is "good" if x is in f(x); x is "bad" if else. Let B be a subset of X which contains the "bad" elements; then B is not in the range of f. Suppose, again for contradiction, B is in the range. Then B = f(x) for some x in X. Is x good? If x is good, then x is in B, which means it's bad! But if x is bad, then x is not in f(x); but then x is in B and thus is good! Contradiction! Thus, B is not in the range of f, and so there is not a bijection between a set and its power set.

Just an interesting general note on power sets and sets. In that vein, is there a relationship between \aleph_0 and \aleph_1?

[Kuroneko]

More generally, the set X and its power set P(X) do not have the same cardinality. Suppose, for contradiction, there is a bijection f:X->P(X) between X and P(X). Now, given an element x in X, x is "good" if x is in f(x); x is "bad" if else. Let B be a subset of X which contains the "bad" elements; then B is not in the range of f. Suppose, again for contradiction, B is in the range. Then B = f(x) for some x in X. Is x good? If x is good, then x is in B, which means it's bad! But if x is bad, then x is not in f(x); but then x is in B and thus is good! Contradiction! Thus, B is not in the range of f, and so there is not a bijection between a set and its power set.

It might be noted that this is the same diagonal cut idea, except that instead of an "nth digit", where n is a natural number, we have an "xth digit", where x is in a completely arbitratry set X, and the allowed digits "values" are "good" and "bad" rather than 0,1,...,b-1.

Just an interesting general note on power sets and sets. In that vein, is there a relationship between \aleph_0 and \aleph_1?

That depends on the set theory one is working in. In ZFC, ℵ1 is the "next greatest" cardinality after ℵ0, in the sense that there is no set X with ℵ_0 < |X| < ℵ1, but nothing stronger than this can be proven or disproven. In ZF+GCH, ℵ1 = 2^{ℵ0}. In pure ZF set theory, the question doesn't even make sense, as there are no alephs since cardinalities are not well-orderable.

[Ariphaos]

Yes, although the sense that there are "clearly more" is rather suspect.

Well, in the intuitive sense, especially after you prove that there are an infinite number of rational numbers in between any two rational numbers.

Ah, no. First, there is no such thing as a set of all sets in any extension of ZF (the standard set theory), ZFC included (one needs the axiom of choice to order the cardinalities as alephs). There's a "class" of all sets in set theories that can talk about such things (e.g., vNBG set theory), but it does not have a cardinality at all. If one takes the generalized continuum hypothesis as an axiom, then an example of an aleph-3 set is P³(N) = P(P(P(N))), where P() is the power set of and Z is the set of natural numbers. Note that even without the generalized continuum hypothesis |P³(N)|≥aleph-3.

I think I got my wording confused and meant to say 'set of all subsets of' ... though I was unaware that you could take power sets of power sets - I didn't delve very deeply into set theory in my math courses. Is that type of thing actually useful in some way?

[Ford Prefect]

A friend asked the following after I showed him this thread:

but if you can state that something has 'more' than something else you have to assume that both things end eventually even if that end can't be measured so it's not true infinity. I'm not going to get past the fact that Infinity is a concept applied to describe the occurrence of something that has no beginning or end if something has no beginning or end it cannot be more so 'endless' than something else that has no beginning or end

Naturally I'm not up on such complex maths as this, so I can't answer him.

[Kuroneko]

Well, in the intuitive sense, especially after you prove that there are an infinite number of rational numbers in between any two rational numbers.

And that's why it is suspect--intuitions are not so easily shared. Personally, I find the one-to-one correspondence "pairing up" as to be the most intuitive definition. (But I have a friend who routinely tries to build a self-consistent replacement because he sees it as some sort of monster.)

I think I got my wording confused and meant to say 'set of all subsets of' ... though I was unaware that you could take power sets of power sets - I didn't delve very deeply into set theory in my math courses. Is that type of thing actually useful in some way?

Well, why not? The power set operation takes any set whatsoever and gives back a set. One can even define ℵω that is larger than any of {ℵ0, ℵ1, ℵ2, ... }, and so on for ℵω+1, ℵω², ..., etc. Large cardinals have their own research area, and then there are fields where the things studied do form proper classes that are "too large" for any set (and thus can be thought of as having cardinality greater than any aleph). A good example of this are surreal numbers, which have important applications in game theory.

[Kuroneko]

but if you can state that something has 'more' than something else you have to assume that both things end eventually even if that end can't be measured so it's not true infinity.

No, you do not. For your proposal to even make sense, there must be some higher structure to the set of considerations. What is a "beginning" of an arbitrary set? To make sense of such things, one must assign the elements of the set some order. But this is not unique, so your proposal is ill-defined.

I'm not going to get past the fact that Infinity is a concept applied to describe the occurrence of something that has no beginning or end if something has no beginning or end it cannot be more so 'endless' than something else that has no beginning or end

What it means for a set to be infinite has a very exact definition of set theory. There are actually several possible definitions that are logically equivalent. One of the more concise ones is "X is infinite iff there is a proper subset of X equinumerous with X." But another is to define finite as: Let A_n = {1,2,3,...,n}, e.g., A_3 = {1,2,3}; a set X is finite iff there exists an n such that A_n is equinumerous with X [in the sense of there being a one-to-one correspondence]. For example, the set {apple,cat,tree} has a correspondence with {1,2,3} (many possible ones, in fact), so it is finite (and obviously has cardinality 3). From there on, define "infinite" as "not finite."

[Ariphaos]

A friend asked the following after I showed him this thread:

but if you can state that something has 'more' than something else you have to assume that both things end eventually even if that end can't be measured so it's not true infinity. I'm not going to get past the fact that Infinity is a concept applied to describe the occurrence of something that has no beginning or end if something has no beginning or end it cannot be more so 'endless' than something else that has no beginning or end

Naturally I'm not up on such complex maths as this, so I can't answer him.

Imagine an infinite line. It has an infinite number of points, and an infinite number of points between any two points along the line.

It is still restricted, however. You can only move in one dimension along a line. Your friend would be correct in stating that an infinite plane is no more 'endless', it just gives you a single additional movement option.

Make it an infinite space, infinite spacetime ... eventually, you can extrapolate that there may be infinite dimensions. -THAT- is a greater infinity, because there is no way you can put lines together to map such an infinity.

[Kuroneko]

Make it an infinite space, infinite spacetime ... eventually, you can extrapolate that there may be infinite dimensions. -THAT- is a greater infinity, because there is no way you can put lines together to map such an infinity.

If you mean that the infinite-dimensional space R^ω = {(x_1,x_2,x_3,...): x_n real} has a greater cardinality than the set of real numbers R, then you are incorrect. If there are only countable many coordinates in the inifinite-dimensional real space, then creating a bijection between it and the reals is a rather easy exercise. Or, if you prefer, (ℵ1)^(ℵ0) = ℵ1.

[Ariphaos]

And that's why it is suspect--intuitions are not so easily shared. Personally, I find the one-to-one correspondence "pairing up" as to be the most intuitive definition. (But I have a friend who routinely tries to build a self-consistent replacement because he sees it as some sort of monster.)

Hah. Well I tend to be of two minds on it. Thinking in terms of programming requires a one-to-one correspondance type of mentality because your resources are ultimately limited, and no matter how much you cheat there is an end to 'numbers'. In that sense I tend to think of it as the better answer, but I still perceive rational numbers to have a greater range or whathaveyou.

Well, why not? The power set operation takes any set whatsoever and gives back a set. One can even define ℵω that is larger than any of {ℵ0, ℵ1, ℵ2, ... }, and so on for ℵω+1, ℵω², ..., etc. Large cardinals have their own research area, and then there are fields where the things studied do form proper classes that are "too large" for any set (and thus can be thought of as having cardinality greater than any aleph). A good example of this are surreal numbers, which have important applications in game theory.

That must be part of the '20th century math' my analysis teacher spoke of. :shock:

Speaking of surreals and such things, can you give a rundown of how math with 'infinite' numbers works at all? Infinitesimals make sense - no one makes it through calculus without some idea of them... but Google does not provide a clear path to understanding them.

[Surlethe]

A friend asked the following after I showed him this thread:

but if you can state that something has 'more' than something else you have to assume that both things end eventually even if that end can't be measured so it's not true infinity. I'm not going to get past the fact that Infinity is a concept applied to describe the occurrence of something that has no beginning or end if something has no beginning or end it cannot be more so 'endless' than something else that has no beginning or end

Naturally I'm not up on such complex maths as this, so I can't answer him.

Did he read the proofs presented in the thread? He's going by the intuitive notion of 'infinite', and, dealing with things like infinite sets, intuition is notoriously misleading. Furthermore, he's wrong about infinity describing something with "no beginning or end"; the whole numbers, for example, certainly have a beginning but have no end: {1, 2, 3, 4, 5, 6, 7, . . . }.

As a tangential point of interest, it's possible to choose a finite length on the real line which has neither beginning nor end: it's called an open interval, and it's defined as (a, b) := {x| a < x < b, x real}.

[Surlethe]

Speaking of surreals and such things, can you give a rundown of how math with 'infinite' numbers works at all? Infinitesimals make sense - no one makes it through calculus without some idea of them... but Google does not provide a clear path to understanding them.

What do you mean "math with infinite numbers"? Do you mean, as in dealing with limits at infinity, or actual operations on infinite sets? Or something else?

[Ariphaos]

If you mean that the infinite-dimensional space R^ω = {(x_1,x_2,x_3,...): x_n real} has a greater cardinality than the set of real numbers R, then you are incorrect. If there are only countable many coordinates in the inifinite-dimensional real space, then creating a bijection between it and the reals is a rather easy exercise. Or, if you prefer, (ℵ1)^(ℵ0) = ℵ1.

It seemed to me that his friend was arguing from a philosophical point of view rather than a mathematical one, so I distinguished between an endless number of possible values for x and an endless number of of x's (x0, x1, x2, x3 ...). In the sense that his friend might see degrees of freedom as a different kind of endless than points on a line. I suppose it was a bad real example, but most people don't believe in noninteger dimensions. I don't even think I do.

[Surlethe]

I suppose it was a bad real example, but most people don't believe in noninteger dimensions. I don't even think I do.

Not to hijack, but ... noninteger dimensions? Is it even possible for that to have meaning?

[Ariphaos]

What do you mean "math with infinite numbers"? Do you mean, as in dealing with limits at infinity, or actual operations on infinite sets? Or something else?

Well it's obviously going to have some relation to limits.

Infinitesimal numbers being used out of derivitives like the 'number' dx, for example, can be shown to have an absolute value less than any positive integer, but it is still a number with meaning. Thus, 1/dx will still have meaning.

I'm not sure if I just answered my own question there :-/

[Ariphaos]

Not to hijack, but ... noninteger dimensions? Is it even possible for that to have meaning?

Kuroneko will probably ream me for this, but AFAIK fractals (at least some of them) are one example. They can't be truly quantified in 2D (say) space, so are considered to be greater than 2D but less than 3D.

Or something like that.

[Surlethe]

Well it's obviously going to have some relation to limits.

Infinitesimal numbers being used out of derivitives like the 'number' dx, for example, can be shown to have an absolute value less than any positive integer, but it is still a number with meaning. Thus, 1/dx will still have meaning.

Less than any positive integer, or less than any positive real? It's not hard to find a number with an absolute value less than 1.

Kuroneko will probably ream me for this, but AFAIK fractals (at least some of them) are one example. They can't be truly quantified in 2D (say) space, so are considered to be greater than 2D but less than 3D.

Or something like that.

Wierd. The dimension of a space is given by the number of independent vectors in the basis; that has to be an integer, though. How could a fractal not be truly quantified in an n-dimensional space?

[Kuroneko]

That must be part of the '20th century math' my analysis teacher spoke of. :shock:

Well, that depends. The aleph concepts are 19th-century, although it wasn't until the 20th that they have received extensive study.

Speaking of surreals and such things, can you give a rundown of how math with 'infinite' numbers works at all?

Well, let's start with something relatively simple--ordinal numbers, which every student of set theory should be familiar with (they have a very intimate connection with the axiom of choice). A set α is an ordinal iff every element β∈α is a subset of α and α is well-ordered by ∈ (a 'well-order' means that any collection of ordinals contains a smallest element, and ∈ is what is used to compare them). That's probably better appreciated by example. Define 0 = ∅ (empty set), 1 = {0} = {∅}, 2 = {0,1} = {∅,{∅}}, 3 = {0,1,2} = {∅,{∅},{∅,{∅}}}, and in general n+1 = n∪{n}. Note that β<α means that α contains β (β∈α). Ordinals have a number of interesting properties. For example, given a set of ordinals Α, the smallest ordinal greater than or equal to all numbers in Α ("supremum", or "sup Α") is the union of all the ordinals in Α (e.g., 1∪2∪3 = 3), and the smallest ordinal contained in it is the intersection (e.g., 2∩3∩4 = 2).

Arithmetic can be defined recursively: α+0 = α, α+1 = α∪{α}, and α+β as the supremum of all {(α+γ)+1: γ<β}. For example, 2+1 = 2∪{2} = {0,1,2} = 3, and 2+2 = sup{(2+0)+1,(2+1)+1} = sup{3,4} = 4, etc. Multiplication is done likewise: α0 = 0 and αβ = sup{αγ+α: γ<β}, e.g., α1 = α0+α = α, α2 = sup{α0+α,α1+α} = α+α, just as expected. Exponentiation: α^0 = 1, α^β = sup{(α^γ)α: γ<β}, e.g., α^1 = sup{(α^0)α} = α, α^2 = sup{(α^0)α, (α^1)α} = αα.

Infinite ordinal numbers come in when one realizes that there no reason to stop with the finite ones. The smallest infinite ordinal is ω = {0,1,2,3,...}. Immediately, ω+1 = {0,1,2,3,...,ω} that is distinct from ω, and so on for ω+n. The ordinal ω2 is the smallest ordinal greater than any of {ω+n}, ω² is the smallest ordinal greater than any of {ωn}, ω³ is the smallest ordinal greater than any of {ω²+n}, and so on to ω^ω and beyond. As an aside, in ZFC it is possible to define a 'cardinal number' as an 'ordinal number not equinumerous to any lesser ordinal'. They have a different arithmetic, however.

Exercise 1: Does 1+ω = ω+1 and does 2ω = ω2?
Exercise 2: Show that the ordinals do not form a set.

Infinitesimals make sense - no one makes it through calculus without some idea of them... but Google does not provide a clear path to understanding them.

Surreal numbers are a very rich field; I can introduce them if you like, but be aware that they are probably too diffucult to get accustomed to in one sitting. The surreal numbers are nicer than ordinals because they form a field (i.e., their arithmetic operations preserve all of the properties of real numbers, such as having multiplicative inverses, which ordinals and cardinals lack).

Not to hijack, but ... noninteger dimensions? Is it even possible for that to have meaning?

Well, infinite cardinalities are not considered integers, this is even trivial.

Kuroneko will probably ream me for this, but AFAIK fractals (at least some of them) are one example. They can't be truly quantified in 2D (say) space, so are considered to be greater than 2D but less than 3D.

Yes, if one considers the similarity dimension or other definitions.

Wierd. The dimension of a space is given by the number of independent vectors in the basis; that has to be an integer, though. How could a fractal not be truly quantified in an n-dimensional space?

There are several different notions of dimensionality. Usually, one works in some sort of Euclidean space (or, at least, something locally homeomorphic to Euclidean space) with the standard Lebesgue measure (one with a basis of open balls). For a general metric space, however, there is quite a bit of freedom in defining the measure. For Hausdorff measure, dimensionality of a space can be fractional.

If you find that counterintuitive, well, there's also fractional calculus (quick, what's the 1/2th derivative of a constant?).

[Ariphaos]

Less than any positive integer, or less than any positive real? It's not hard to find a number with an absolute value less than 1.

Err, real, right. Easiest way to think of it is those infinitesimal slices you end up 'making' in calculus. Then take the reciprical.

Wierd. The dimension of a space is given by the number of independent vectors in the basis; that has to be an integer, though. How could a fractal not be truly quantified in an n-dimensional space?

Think of it this way. If you put your pencil down, and started drawing, you could not finish. The best you can do, no matter how much paper and time you are given, is to make an approximation. The full description possesses what is called a capacity dimension, IIRC, which is a real number. I am by no means qualified to give the full answer as I'd just muddy it.

Another (unrelated) way of thinking of noninteger dimensions is your perception of them. For example, assuming you can see and aren't reading this through braile, and have two eyes, you do not possess full, 3-dimensional vision, but rather somewhere better than 2D (you have the benefit of parallax) but not true 3D (you do not see all aspects of a given 3D space in front of you).

[Surlethe]

Well, infinite cardinalities are not considered integers, this is even trivial.

My imprecision.

There are several different notions of dimensionality. Usually, one works in some sort of Euclidean space (or, at least, something locally homeomorphic to Euclidean space) with the standard Lebesgue measure (one with a basis of open balls). For a general metric space, however, there is quite a bit of freedom in defining the measure. For Hausdorff measure, dimensionality of a space can be fractional.

Okay. I've only worked with Euclidean spaces thus far in my education, so I see now where my (admittedly yet small) education lacks.

If you find that counterintuitive, well, there's also fractional calculus (quick, what's the 1/2th derivative of a constant?).

0?

[Surlethe]

Wierd. The dimension of a space is given by the number of independent vectors in the basis; that has to be an integer, though. How could a fractal not be truly quantified in an n-dimensional space?

Think of it this way. If you put your pencil down, and started drawing, you could not finish. The best you can do, no matter how much paper and time you are given, is to make an approximation. The full description possesses what is called a capacity dimension, IIRC, which is a real number. I am by no means qualified to give the full answer as I'd just muddy it.

Another (unrelated) way of thinking of noninteger dimensions is your perception of them. For example, assuming you can see and aren't reading this through braile, and have two eyes, you do not possess full, 3-dimensional vision, but rather somewhere better than 2D (you have the benefit of parallax) but not true 3D (you do not see all aspects of a given 3D space in front of you).

That's helpful. Thanks.