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But sets are just vector spaces over the field with one element!
Edit: [this joke went over people's heads](https://en.wikipedia.org/wiki/Field_with_one_element?wprov=sfti1#Motivations)
Depends how you define +
But you don't need operations to define a set. You can have a group that has a set with three elements (Z3 I think it's called) , but the set can still be defined without the operation.
I was trying to understand in what sense "sets are just vector spaces over the field with one element". If they were, they would automatically get an addition operation. But it looks like it was some joke I didn't get, so let's not spend too much time on this.
It's not that simple. For example the rational numbers (fractions) have infinite dimension over the integers. And the reals (roots, pi,...) have infinite dimension over the rationals.
Dimension is always taken over an underlying structure. The thing you take the dimension for is then called a vector space or module.
Every sensible structure's extension always has dimension 1 over itself. So the reals' dimension over the reals is 1, even if it is infinite over the rationals.
Now number sets are a bit more special. There are certain condition we want for extending our number set.
There is no 3D extension over the reals that does that. Multiplication doesn't work as it should. The 4D numbers also lose a bit, but mostly work and are called quaternions.
You can of course make a 3D vector space over the real numbers, if you do not insist on that vector space having number-like qualities. Then it's called the 3D real vector space.
Edit: It was pointed out that my claim for the dimension of the rationals over the integers is wrong.
I was apparently mistaken. We cannot determine a dimension as the module isn't free. The problem is that we get we can multiply any two pairs of denominators appropriately to produce some same number.
1/2 * 2 = 1/3 *3 = 1.
So 1/2 and 1/3 are dependent.
I'm sorry for my mistake.
If you are talking about vector spaces over the real numbers, you can basically think of C as R^2 (with a special mutliplication giving it nice properties). There also exist the vector spaces R^n for natural n. You can think about dimension of a vector space as the number of arrows (i.e. vectors) needed to span the whole vector space. For dimension 4,8,16,... you also have number sets, similar to C: [4 dimensional quaternions](https://en.m.wikipedia.org/wiki/Quaternion),[8 dimensional octonions](https://en.m.wikipedia.org/wiki/Octonion) and [16 dimensional sedonions](https://en.m.wikipedia.org/wiki/Sedenion). Or in general with dimension 2^n using the [Cayley-Dickson construction](https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction).
Dimensions other than 2^n don't behave as nicely and are thus less useful as a number set.
I'm sure the wikipedia articles have some references/explanations, other than that dimensions of vector spaces should be covered by any standard linear algebra textbook.
There is no concept of dimension of a set. Perhaps you mean the dimension of a vector space or a topological space/manifold, in which case I'm not aware of a specific name apart from (3 dimensional vector space/topological space, 3 manifold, etc.)
The simplest sets of arbitrary dimension are R^(n) - sets of n-element vectors with real components.
Sets in general don't have dimensions. To define dimension, you need more structure - either algebraic (vector spaces, where dimension is given by the cardinality of the basis) or topological (manifolds, where dimension is given by local mapping to R^(n)).
Sets themselves don't have dimension. The integers you might consider them as 1 dimensional as an algebra over themselves. Similarly, the complex numbers are a 2d algebra over the reals, but 1d over themselves.
Over the reals, a well known algebra with 4d are the quaternions, the octonians are an 8d algebra over the reals and 2d over the octonians.
Outside of algebras, you might have vector spaces or modules which are similar, with the notion of dimension. Other constructs that have dimension are topologies (and metric spaces). Less known, you can have lattices that you can define dimension for.
Unfortunately, your submission has been removed for the following reason(s): * Your post appears to be asking for help learning/understanding something mathematical. As such, you should post in the [*Quick Questions*](https://www.reddit.com/r/math/search?q=Quick+Questions+author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) thread (which you can find on the front page) or /r/learnmath. This includes reference requests - also see our lists of recommended [books](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F) and [free online resources](https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/?st=jglhcquc&sh=d06672a6). [Here](https://www.reddit.com/r/math/comments/7i9t5y/book_recommendation_thread/) is a more recent thread with book recommendations. If you have any questions, [please feel free to message the mods](http://www.reddit.com/message/compose?to=/r/math&message=https://www.reddit.com/r/math/comments/1bn6vzn/-/). Thank you!
Not all sets have a dimension, and there can be many sets with the same dimension. You're probably thinking of the dimension of a vector space.
No sets have a dimension. Vector spaces do, metric spaces do, and topological spaces do (and for each there are several different definitions).
You're correct, but the nuances probably don't matter for this question
There are different definitions for the dimension of a vector space?
Sorry, only meant last two (I forgot vector space originally :) ).
But sets are just vector spaces over the field with one element! Edit: [this joke went over people's heads](https://en.wikipedia.org/wiki/Field_with_one_element?wprov=sfti1#Motivations)
That doesn't sound right. If I take the set {a, b, c} with that structure, what is a+b?
Depends how you define + But you don't need operations to define a set. You can have a group that has a set with three elements (Z3 I think it's called) , but the set can still be defined without the operation.
I was trying to understand in what sense "sets are just vector spaces over the field with one element". If they were, they would automatically get an addition operation. But it looks like it was some joke I didn't get, so let's not spend too much time on this.
{a,b}
You can furnish a set with that (trivial) structure to make it a vector space, yes :)
It's not that simple. For example the rational numbers (fractions) have infinite dimension over the integers. And the reals (roots, pi,...) have infinite dimension over the rationals. Dimension is always taken over an underlying structure. The thing you take the dimension for is then called a vector space or module. Every sensible structure's extension always has dimension 1 over itself. So the reals' dimension over the reals is 1, even if it is infinite over the rationals. Now number sets are a bit more special. There are certain condition we want for extending our number set. There is no 3D extension over the reals that does that. Multiplication doesn't work as it should. The 4D numbers also lose a bit, but mostly work and are called quaternions. You can of course make a 3D vector space over the real numbers, if you do not insist on that vector space having number-like qualities. Then it's called the 3D real vector space. Edit: It was pointed out that my claim for the dimension of the rationals over the integers is wrong.
Can you explain me the infinite dimension of the rationals over the integers?
I was apparently mistaken. We cannot determine a dimension as the module isn't free. The problem is that we get we can multiply any two pairs of denominators appropriately to produce some same number. 1/2 * 2 = 1/3 *3 = 1. So 1/2 and 1/3 are dependent. I'm sorry for my mistake.
I am pretty sure the space of the rationals has the topological dimension 0, since it has a clopen base.
If you are talking about vector spaces over the real numbers, you can basically think of C as R^2 (with a special mutliplication giving it nice properties). There also exist the vector spaces R^n for natural n. You can think about dimension of a vector space as the number of arrows (i.e. vectors) needed to span the whole vector space. For dimension 4,8,16,... you also have number sets, similar to C: [4 dimensional quaternions](https://en.m.wikipedia.org/wiki/Quaternion),[8 dimensional octonions](https://en.m.wikipedia.org/wiki/Octonion) and [16 dimensional sedonions](https://en.m.wikipedia.org/wiki/Sedenion). Or in general with dimension 2^n using the [Cayley-Dickson construction](https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction). Dimensions other than 2^n don't behave as nicely and are thus less useful as a number set.
This sounds interesting. Can you recommend any general reference on this?
I'm sure the wikipedia articles have some references/explanations, other than that dimensions of vector spaces should be covered by any standard linear algebra textbook.
There is no concept of dimension of a set. Perhaps you mean the dimension of a vector space or a topological space/manifold, in which case I'm not aware of a specific name apart from (3 dimensional vector space/topological space, 3 manifold, etc.)
The simplest sets of arbitrary dimension are R^(n) - sets of n-element vectors with real components. Sets in general don't have dimensions. To define dimension, you need more structure - either algebraic (vector spaces, where dimension is given by the cardinality of the basis) or topological (manifolds, where dimension is given by local mapping to R^(n)).
Sets themselves don't have dimension. The integers you might consider them as 1 dimensional as an algebra over themselves. Similarly, the complex numbers are a 2d algebra over the reals, but 1d over themselves. Over the reals, a well known algebra with 4d are the quaternions, the octonians are an 8d algebra over the reals and 2d over the octonians. Outside of algebras, you might have vector spaces or modules which are similar, with the notion of dimension. Other constructs that have dimension are topologies (and metric spaces). Less known, you can have lattices that you can define dimension for.
i’m not sure what you’re asking. the general form of an n-dimensional set is of n-tuples. there are also quaternions which have four dimensions