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Consider the set of all infinite arithmetic sequences of real numbers:

$$\forall f,d\in\mathbb R\ (f,f+d,f+2d,\cdots,f+id,\cdots)$$

Most people would say that the set of all of these sequences is parameterized by $f$ and $d$ thus there is a mapping from $\mathbb R^2$ to these sequences. Even further people would say that, since this is the lowest number of parameters possible, the dimension of the space of sequences is $2$.

Now consider everything I said above but for integers. It would still take at minimum $2$ integers to parameterize it.

But what about space-filling curves? A space-filling curve can map the real line to the real plane, meaning only $1$ variable is needed to parameterize the plane. Does that mean the plane is $1$ dimensional? You might say the parametrization has to be continuous (something the space filling curve is not) but then what about the integer sequence case? Can that case not even be parametrized because the integers are not continuous?

And in general, if two sets have the same cardinality then there must only be 1 variable needed two parameterize one to the other, no? The definition of cardinality is that there exists some bijective function from one to the other. Or equivalently we could consider the set of pairs $(f,d)$ to map to the sequences rather than the two individual variables.

I've been trying to pick this apart for a while now since parametrizations are everywhere and, especially in my calculus class, we talk of the dimension of objects as being associated with the minimum number of parameters needed to paramaterize it (i.e a sphere while embedded in 3-space is actually 2-dimensional).

In that case it might be as simple as an implication that the parameters be differentiable since we are doing calculus but that fact that this is never addressed has me confused.

But in general, is parameterize a term that doesn't really mean anything concrete? Does it change meaning depending on the type of math you're doing? What if these two definitions collide like with the sequences?

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  • $\begingroup$ Space filling curves are continuous, but wildly non-differentiable. There are a number of different definitions of "dimension" but for the one you're seeing in calculus, the functions have to be differentiable. As to the statement that the "space" of infinite arithmetic sequences of real numbers is two-dimensional, I question that. What kind of a "space" is it? It's not a vector space, for example. Without knowing that, we can't even begin to decide what definition of dimension to use. $\endgroup$ – saulspatz Oct 17 '18 at 17:07
  • $\begingroup$ So I guess dimension relies on a complete vector space? But disregarding dimensions entirely where do parameters fit into this? $\endgroup$ – Ozaner Hansha Oct 17 '18 at 18:16
  • $\begingroup$ No, I didn't say that. There is a definition of dimension of a vector space. Other sorts of spaces have a concept dimension, too. To say that a set is parameterized by a function just means that it is the range of that function. Parameterization is very simple. It's dimension that's causing your difficulties. $\endgroup$ – saulspatz Oct 17 '18 at 18:19
  • $\begingroup$ But why do people say 'minimum' amount of parameters when the answer is always 1 (assuming there is some function that can map between them). I just had a math exam that said give the minimum amount of parameters used to parameterize the set of finite arithmetic sequences. I got it right and said 3 parameters (first term, difference, and length) but I'm thinking I should be wrong since I could technically think of a mapping that only used one number. $\endgroup$ – Ozaner Hansha Oct 17 '18 at 18:37
  • $\begingroup$ When you're talking about calculus, there are restrictions on the nature of the functions. As to what you say about arithmetic sequences, I agree, although I think you will admit there isn't a very natural mapping that uses only one number. All I can say is that I think the question is rather informal; the instructor wanted you to show that you understood the idea of parameterization. I suppose we could change it to something like, "What is the smallest number of variable one can use to 'naturally' parameterize the set of arithmetic sequences," which is not strictly mathematical, either. $\endgroup$ – saulspatz Oct 17 '18 at 18:47

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