Question about the FAQ

[Palps]

Hi, this thread will deal with a question I had while re-reading the wiki FAQ.
So, first, you assume here that an n-dimensional being/object is composed of an infinity of n-1-dimensional objects, like a two-dimensional square composed of an infinity of one-dimensional segments.
And then, a few questions later, you assume here that a set A composed of an infinity of points of dimension 0 was itself a set of dimension 0.
My question is about this contradiction: if an arbitrary object (let's think of it as a set) of dimension n is composed of an infinity of objects of dimension less than it, the set A (which I mentioned before) that contains an infinity of objects of dimension 0 should logically itself be a set of dimension 1, since if we think of dimension 0 as dimension n-1 (dimension n was dimension 1), an infinity of objects of dimension n-1 makes the set an n-dimensional set (1), as I clarified before.
Either I have misinterpreted or it would be contradictory, I would like your opinion.
(And I would like to apologize in advance if I don't respect some of the thread rules, I'm new and frankly don't know enough about it.)
Palps ~

 

[Cat275]

Not exactly sure what you mean, but from what i'm seeing your implying that the tiering FAQ says that a set A that has a infinite amount of 0 is a 1 dimensional object. But later it was stated that a set A that has infinite amount of 0 dimensions is 0 dimensional and thus a contradiction.

Well in simple terms, a set A is 1 dimensional when there are a uncountable infinite amount of 0 dimensional objects.

(Quoting:

In a way, yes, though not how most would think when using this word. Basically, an arbitrary object of dimension n is essentially comprised by the total sum of uncountably infinite objects of one dimension less)

Which means you need atleast a 2^Aleph 0 amount of 0 dimensional objects by the continuum hypothesis (or the real number line, it could also be portrayed as something that has a aleph 0 amount of ^ notations depending on how you view and interpret an epsilon number) to be 1d.

But bassically a infinite 0 dimensional object does not create a a new dimensional segment and instead you need a uncountable infinite of a n-dimensional object (in this case n=0) or a closed set of 0-1 ([0,1], more generally known as the unit interval.) to satisfy a new segment of dimensions.

 

[Palps]

Not exactly sure what you mean, but from what i'm seeing, your implying that the tiering faq says that a set A that has a infinite amount of 0 is a 1 dimensional object and later it was said that a set A that has infinite amount of 0 dimensions is 0 dimensional and thus a contradiction.

Well in simple terms, a set A is 1 dimensional when there are a uncountable infinite amount of 0 dimensional objects.

(Quoting:

In a way, yes, though not how most would think when using this word. Basically, an arbitrary object of dimension n is essentially comprised by the total sum of uncountably infinite objects of one dimension less)

Which means you need atleast a 2^Aleph 0 amount of 0 dimensional objects by the continuum hypothesis (or the real number line, it could also be portrayed as something that has a aleph 0 amount of ^ notations depending on how you view an epsilon number) to be 1d.

But bassically a infinite 0 dimensional object does not create a a new dimensional segment and instead you need a uncountable infinite of a n-dimensional object (in this case n=0) or a closed set of 0-1 ([0,1], more generally known as the unit interval.) to satisfy a new segment of dimensions.

What I mean, in simple terms, is that the wiki assumes that a set, which we will call X, is a set of dimension n only and only if it is composed of an incalculable (infinite) number of arbitrary objects of dimension n-1. So, if we consider that n = 3, the set/object X that we consider would be a cube, but since it is composed of an incalculable number of arbitrary objects of dimension n-1, thus 2, the object X contains an infinity of two-dimensional squares. Now, in the case of the set having an infinity of 0-dimensional points, since we have previously defined an n-dimensional object as being composed of an infinity of lower-dimensional objects, the set should logically be 1D, and not remain 0-dimensional, because the reasoning applies here too: a 1D set, composed of an infinity of 0D points, which defines our case well.

 

[Cat275]

What I mean, in simple terms, is that the wiki assumes that a set, which we will call X, is a set of dimension n only and only if it is composed of an incalculable (infinite) number of arbitrary objects of dimension n-1. So, if we consider that n = 3, the set/object X that we consider would be a cube, but since it is composed of an incalculable number of arbitrary objects of dimension n-1, thus 2, the object X contains an infinity of two-dimensional squares. Now, in the case of the set having an infinity of 0-dimensional points, since we have previously defined an n-dimensional object as being composed of an infinity of lower-dimensional objects, the set should logically be 1D, and not remain 0-dimensional, because the reasoning applies here too: a 1D set, composed of an infinity of 0D points, which defines our case well.

Not exactly sure how that's the case but in mathematics we have a thing called real coordinate space which measures the n tuples of real numbers, an example of this is a space correspondance of Rx² is 2 dimensional because it corresponds to a 2 dimensional object where there is a closed set of 1-2 and 0-1 or a multiplication of [0,1]'s (unit interval) which creates a intersection of a 1 dimensional objects to satisfy a 2 dimensional object, but bassically the wiki uses and extrapolate this "mathematic" called real coordinate space to scale dimensions.

(Or geometric dimensions in general, etc.)

That aside, the idea of infinite 0 dimensional objects = 1 dimensional is like a Nx¹ in which doesn't correspond to a real number amount of 1 n tuple and doesn't create 1d in real coordinate space.

(Which is why a set X is still 0 dimensional when it has infinite 0 dimensional sets, also i'm pretty sure the points needed for dimensions increases as they intersect.)

 

[Palps]

Not exactly sure how that's the case but in mathematics we have a thing called real coordinate space which measures the n tuples of real numbers, an example of this is a space correspondance of Rx² is 2 dimensional because it corresponds to a 2 dimensional object where there is a closed set of 1-2 and 0-1 or a multiplication of [0,1]'s (unit interval) which creates a intersection of a 1 dimensional objects to satisfy a 2 dimensional object, but bassically the wiki uses and extrapolate this "mathematic" called real coordinate space to scale dimensions.

(Or geometric dimensions in general, etc.)

That aside, the idea of infinite 0 dimensional objects = 1 dimensional is like a Nx¹ in which doesn't correspond to a real number amount of 1 n tuple and doesn't create 1d in real coordinate space.

(Which is why a set X is still 0 dimensional when it has infinite 0 dimensional sets, also i'm pretty sure the points needed for dimensions increases as they intersect.)

I'm not sure I understood this idea very well but I will look into it more, thanks again.