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[MagmaMcFry]

Actually, the Hausdorff dimension of a point is 0, and since the 0-dimensional real Hausdorff measure of a set is exactly its cardinality, a single point has "size" 1.

The thing is, at dimension 0 that single point is everything and nothing.

It cannot have a defined dimension because it lacks dimension.

Think of it this way. What is the dimensions of an object without dimensions?

Please, sir, don't just casually dismiss my rad mathsy explanations.

1) A single point is only sometimes everything and never nothing (and dimension has nothing to do with that, although a single point is only ever not a zero set WRT a Lebesgue measure if the dimension of the Lebesgue measure is 0).

2) If a point lacks any dimensions, it has dimension 0, which is very well-defined and consistent.

3) The dimension of an object without dimensions is 0, and the dimensions of an object without dimensions are none.

[Neonivek]

If a point lacks any dimensions, it has dimension 0, which is very well-defined and consistent

This isn't an object with the dimensions of nothing this is an object with "no" dimensions.

If you were to write the dimensions of a 1 dimensional object it would either be a single number or a range of numbers.

You cannot write the dimensions of a 0 dimensional object because it has no dimensions. It exists in a non-dimensional space.

Its dimensions are not defined because you are essentially dividing by zero.

[Max White]

There is a difference between a single point and a zero dimensional plane. For example, on a 1d plane you could have several points existing at (1), (4) and (6), and on a 2d plane you have (1,7), (4, -8), (6, .4), but what about a zero dimensional plane? You end up with (), having zero coordinates, so how does one define size when you have no coordinates?
A zero dimensional array means you have an element that exists with a multiplicity of one. One element exists (Or in some cases, may or may not exist) and it will never have location.

[da_nang]

There is a difference between a single point and a zero dimensional plane. For example, on a 1d plane you could have several points existing at (1), (4) and (6), and on a 2d plane you have (1,7), (4, -8), (6, .4), but what about a zero dimensional plane? You end up with (), having zero coordinates, so how does one define size when you have no coordinates?
A zero dimensional array means you have an element that exists with a multiplicity of one. One element exists (Or in some cases, may or may not exist) and it will never have location.

In other words, a point is an element of a set and not the set itself.

A dimension is a set.

Let Σ be a non-empty set of numbers.
The nth dimension of Σ is thus Σⁿ. The size of Σⁿ is defined as |Σⁿ|= |Σ|ⁿ
Thus the size of the zeroth dimension is |Σ|⁰ = 1

A point on the other hand is just an element. What is the "size" of a point? It has no volume, no area, no length. It can be moved around to any coordinate. Much like a null vector. Perhaps a point is the null vector. Oh sure, you can represent the position of a point with a vector from origin, but the origin can be shifted to place said point at origin and thus we're back to a null vector.

So the size of a point might as well be the size of the null vector. In other words, |0| = 0.

[Neonivek]

It can be moved around to any coordinate

Cannot be moved because that would require it to have definition.

I cannot express this well but one of the reasons it cannot be 1 is because a 0 dimensional object cannot even assert its own existence.

[MagmaMcFry]

If a point lacks any dimensions, it has dimension 0, which is very well-defined and consistent

This isn't an object with the dimensions of nothing this is an object with "no" dimensions.

Okay, what you're doing here is confusing the concept of "dimension" and "dimension", which, to be fair, is quite confusing. In a mathematical sense, "dimension" is a property of a subset of a linear or affine space, and it is normally a non-negative integer which is the minimal amount of axes of any coordinate system that can fully encompass all points in that set. So the dimension of any set S consisting only of a single point P is 0, since there is a coordinate system with 0 axes (with origin at P) that is enough to fully describe all points in S. Let's not go into Hausdorff dimensions for now.

If you were to write the dimensions of a 1 dimensional object it would either be a single number or a range of numbers.

I don't know what you mean. Example please?

You cannot write the dimensions of a 0 dimensional object because it has no dimensions. It exists in a non-dimensional space.

Sure you can, just as you can write the "dimensions" of a 1 dimensional object. And while the point itself is 0-dimensional, the space it is inside is not necessarily 0-dimensional too.

Its dimensions are not defined because you are essentially dividing by zero.

No, dividing by zero does not happen.

Cannot be moved because that would require it to have definition.

It has a perfectly valid definition. Let's use the point (2,3) in the 2-dimensional Euclidean coordinate space. You can translate that to (5,4) without any problems.

I cannot express this well but one of the reasons it cannot be 1 is because a 0 dimensional object cannot even assert its own existence.

Sure it can. This all sounds like you're completely mixing up everything at once. Just because a necessary axis of a 0-dimensional object doesn't exist, doesn't mean that the 0-dimensional object itself does not exist.

[da_nang]

It can be moved around to any coordinate

Cannot be moved because that would require it to have definition.

A point is defined as a zero-dimensional object. It can exist anywhere in any n-space and we define its position with coordinates using a second point as a reference. That reference point can be chosen arbitrarily hence the point can exist at any location.

The dimension is the number of coordinates needed to specify the location of a point in an n-space and the n-space is the set of all possible locations of said point. By this definition, a dimension doesn't have a "size", it's just a number. An n-space does have a "size", which is called cardinality. A point does have a "size", but it's a degenerate case of all n-dimensional measurements where n > 0.
These measurements, if the units are omitted, are all 0. Thus the "size" of a point is 0. Or if we use the null vector to represent a point, its magnitude is 0.

[MagmaMcFry]

It can be moved around to any coordinate

Cannot be moved because that would require it to have definition.

A point is defined as a zero-dimensional object. It can exist anywhere in any n-space and we define its position with coordinates using a second point and a set of axes as a reference. That reference point and those axes can be chosen arbitrarily hence the point can exist at any location.

Fix'd.

The dimension is the number of coordinates needed to specify the location of a point in an n-space and the n-space is the set of all possible locations of said point. By this definition, a dimension doesn't have a "size", it's just a number. An n-space does have a "size", which is called cardinality. A point does have a "size", but it's a degenerate case of all n-dimensional measurements where n > 0.
These measurements, if the units are omitted, are all 0. Thus the "size" of a point is 0. Or if we use the null vector to represent a point, its magnitude is 0.

The k-dimensional measure of the point (where 0 < k <= n) is 0 u^k (where u is any unit), but the 0-dimensional measure of the point is 1 u⁰ = 1.

[Max White]

A dimension is a set.

Very close, but slightly different.
If I say I have 2 sets, I have two separate sets. If I say I have two dimensions, I have one set containing n number of sets.
I guess a dimension is a very specific sort of set? So it would be accurate to say a dimension is a set, but not very precise. Although saying I have 2 dimensions is still saying I have a heck of a lot more sets, so it is tricky.

[da_nang]

A dimension is a set.

Very close, but slightly different.
If I say I have 2 sets, I have two separate sets. If I say I have two dimensions, I have one set containing n number of sets.
I guess a dimension is a very specific sort of set? So it would be accurate to say a dimension is a set, but not very precise. Although saying I have 2 dimensions is still saying I have a heck of a lot more sets, so it is tricky.

Yeah, I changed that in the second post. A dimension n is just a positive integer indicating the number of coordinates needed to represent a point in an n-space. An n-space is a set, but the dimension is a structural property of that set.

It can be moved around to any coordinate

Cannot be moved because that would require it to have definition.

A point is defined as a zero-dimensional object. It can exist anywhere in any n-space and we define its position with coordinates using a second point and a set of axes as a reference. That reference point and those axes can be chosen arbitrarily hence the point can exist at any location.

Fix'd.

I was under the impression that coordinates couldn't exist without axes (or directions if you will) so it was an implicit requirement.

The dimension is the number of coordinates needed to specify the location of a point in an n-space and the n-space is the set of all possible locations of said point. By this definition, a dimension doesn't have a "size", it's just a number. An n-space does have a "size", which is called cardinality. A point does have a "size", but it's a degenerate case of all n-dimensional measurements where n > 0.
These measurements, if the units are omitted, are all 0. Thus the "size" of a point is 0. Or if we use the null vector to represent a point, its magnitude is 0.

The k-dimensional measure of the point (where 0 < k <= n) is 0 u^k (where u is any unit), but the 0-dimensional measure of the point is 1 u⁰ = 1.

... Hence why I left out n=0. Those special cases, man...

EDIT: Wait, I mean RRRAAAGGGEEE SPECIAL CASES SUCKS!!!111!‼‼‼ LACK OF RAGE IN RAGE THREAD‼‼‼

[Max White]

Except it need not be a positive integer. Well at least taking some definitions of the word. Video games are a great example, because they are some what self contained universes with two or three dimensions that use nice, simple euclidean, Newtonian physics. Anyway it is very easy to be at a negative or a decimal, and these still count as dimensional places.

Although this isn't always true. In software arrays you are limited to positive integers, but that doesn't make it the rule for all things thought of as a dimension.

[MagmaMcFry]

It can be moved around to any coordinate

Cannot be moved because that would require it to have definition.

A point is defined as a zero-dimensional object. It can exist anywhere in any n-space and we define its position with coordinates using a second point and a set of axes as a reference. That reference point and those axes can be chosen arbitrarily hence the point can exist at any location.

Fix'd.

I was under the impression that coordinates couldn't exist without axes (or directions if you will) so it was an implicit requirement.

But then the reference point would be an implicit requirement too? Also I'm not clarifying that for you.

The dimension is the number of coordinates needed to specify the location of a point in an n-space and the n-space is the set of all possible locations of said point. By this definition, a dimension doesn't have a "size", it's just a number. An n-space does have a "size", which is called cardinality. A point does have a "size", but it's a degenerate case of all n-dimensional measurements where n > 0.
These measurements, if the units are omitted, are all 0. Thus the "size" of a point is 0. Or if we use the null vector to represent a point, its magnitude is 0.

The k-dimensional measure of the point (where 0 < k <= n) is 0 u^k (where u is any unit), but the 0-dimensional measure of the point is 1 u⁰ = 1.

... Hence why I left out n=0. Those special cases, man...

But that special case is precisely what we're talking about!

Except it need not be a positive integer. Well at least taking some definitions of the word. Video games are a great example, because they are some what self contained universes with two or three dimensions that use nice, simple euclidean, Newtonian physics. Anyway it is very easy to be at a negative or a decimal, and these still count as dimensional places.

Although this isn't always true. In software arrays you are limited to positive integers, but that doesn't make it the rule for all things thought of as a dimension.

I guess it isn't the rule for all things thought of as a dimension, but technically what you're talking about here is the "size" of a software array. Also, negative dimensions?

[Max White]

Dimensions can't be negative, but things can exist at a negative point along an axis. Unless if Nang meant that, and I just misinterpreted. And that seems very likely. Do I feel silly? A little, but you live and learn.

[da_nang]

Except it need not be a positive integer. Well at least taking some definitions of the word. Video games are a great example, because they are some what self contained universes with two or three dimensions that use nice, simple euclidean, Newtonian physics. Anyway it is very easy to be at a negative or a decimal, and these still count as dimensional places.

Although this isn't always true. In software arrays you are limited to positive integers, but that doesn't make it the rule for all things thought of as a dimension.

Wait, what? How can you have negative number of coordinates? That doesn't make any sense. You can have 0, 1, 2, 3,..., n coordinates, but -1, -2, -3,..., -m coordinates? Or even noninteger number of coordinates. That's like saying you have one and a half man (not including Tyrion). I think you're confusing number of coordinates with the value of a coordinate.

Like in 3-dimensional space, you need 3 coordinates to represent a point in that space. The space has a dimension of 3. But let's say that space is R³, then a coordinate can take any number k in R as its value.

[MagmaMcFry]

Except it need not be a positive integer. Well at least taking some definitions of the word. Video games are a great example, because they are some what self contained universes with two or three dimensions that use nice, simple euclidean, Newtonian physics. Anyway it is very easy to be at a negative or a decimal, and these still count as dimensional places.

Although this isn't always true. In software arrays you are limited to positive integers, but that doesn't make it the rule for all things thought of as a dimension.

Wait, what? How can you have negative number of coordinates? That doesn't make any sense. You can have 0, 1, 2, 3,..., n coordinates, but -1, -2, -3,..., -m coordinates? Or even noninteger number of coordinates. That's like saying you have one and a half man (not including Tyrion). I think you're confusing number of coordinates with the value of a coordinate.

Like in 3-dimensional space, you need 3 coordinates to represent a point in that space. The space has a dimension of 3. But let's say that dimensional is R³, then a coordinate can take any number k in R as its value.

Actually, you can have non-integer dimensions with an extended definition of dimension, namely the Hausdorff dimension definition, which is defined for every metric space. For example, the dimension of the Koch snowflake is log(4)/log(3).