Well, for one, it's well-defined, so it passes the hurdle where the infinite circle idea seems to fall flat.
Probably something along the lines of taking the set of all points that have two real numbers as their orthogonal coordinates, or something along those lines.
Again, you seem to be laboring under severe misconceptions about limit taking. You didn't start with a finite plane that you extended to infinity, so no, you didn't take a limit here.
No I'm extending the circle indefinitely, without limit. I don't need a function that relates the radius to the curvature. All I need is indefinite extension (hell, I don't even need infinity, because that's implicit).
And you seem to be unable to distinguish between the modern concept in calculus, of the limit of a function, and the ancient Greek concept of indefinite extension.
I didn't start with a finite plane, because I can't--the plane is infinite.
More misconceptions. Please look up what "taking the limit to infinity" means.
Actually, you already have set that up in your OP. That you didn't realize that doesn't mean you didn't do it.
Which is exactly what limit taking is...
If you don't need infinity, you can't have an infinite circle. Are you now abandoning your infinite circle claims completely?
I don't know; "circle of directions" is something you came up with, so why don't you answer your own question?
Well, there you go, I guess?
I have no clue what a "circle of directions" is, so I can't provide you with any answers regarding that.
Oh bullshit.
Since I understand that limit taking can be seen as an generalization of indefinite extension, I'm perfectly capable of distinguishing the two, thank you very much.
So why were you talking about limit taking there then?
I suppose, yes.
Wrong. It's "of the infinite plane". I'm constructing it; that was the task you asked of me. One cannot construct a thing inside the thing that one is trying to construct; that's ridiculous.
What do you mean, "where"? It's not like I'm placing points into an already existing space. In fact, the origin is an arbitrary choice, so I guess you could answer "anywhere you want"?
How can you have an infinite plane when you are still busy constructing it? You are making no sense.
What is a "circle of directions"?
The more I read "circle of directions", the less coherent the term becomes. How can a circle be constructed out of directions?
Please provide that proof then! I've asked you multiple times for proof now, and you continue to refuse while also continuing to claim the proof exists. It's approaching intellectually dishonesty, to be frank.
A circle being located at infinity doesn't mean the circle is infinite in size, so I don't know why you are bringing this up?
Well, obviously.
360 degrees, obviously.
Yes, because if you read the definition of circle I gave earlier, this obviously isn't one.
Actually, not "no".
Why do you feel the need to start defining things that are irrelevant to the discussion at hand? Are you trying to pull this thread off-topic?
There's a difference between "in the infinite plane", and "of the infinite plane"? That's pretty cute.
Ha ha hah . . . etc
You aren't very well-read, are you? A circle of directions isn't that common a term, but it is well understood (and not really hard to understand, unless you're too cute to be bothered).
Why should I accept what you say about relevance? Who TF are you?
Post #59: You: "How can the infinite plane exist? How do you construct it?"
That's a response to one of your posts; you've taken this out of context so you can troll this thread.
But the diameter of the circle also becomes infinite, no? Why not an infinitely large curved plane?
A circle with an infinite radius would also have an infinite diameter, yes.
That would be non-Euclidean; I've explicitly stayed away from that in this thread.
That's what the whole discussion between arfa brane and me was about. It seems that both of us have come up with arguments why infinite circles can't exist, without having found any counter-arguments, so it seems that if such a thing is possible, then yes, it ceases to be a circle.
It seems that it's a matter of choice whether an infinite circle is still a circle; some say no it can't be, by definition, some say it is a circle "at infinity", but loses most of the properties of a circle.NotEinstein said:
So, what happens to the plane, it also disappears, no?
What plane do you mean, exactly?
I don't think there's any 2D plane changing into a 1D line in this situation?
Are you "inflating"/expanding the space itself, or things within the space?
Why would a 3D space change into 1D lines in all directions?
How is this related to anything? Also, are you suggesting you know whether the universe is infinite or not (in spatial size)?
No, because we are talking about abstract concepts, not physical reality.
A circle forms a 2 D plane, no? All points equi-distant from the center?
Is the center of the circle not in the center of the plane as well as in the center of a sphere?
That would be an example of a 3D sphere inflated infinitely. If the surface expands infinitely, does the sphere cease to be sphere at some point?
Does a sphere not have an infinite number of circumferences, creating a "volume"? Unlike a circle which has just one circumference, creating a "circle"?
No, I'm posing the question. It seems related to the OP and to the current example of a circle, no?
The argument would still need to be logically sound, no?
That's why I am asking....
A circle lies in a plane, but it isn't one itself. Are you perhaps thinking of a disc?
Not of the plane the circle is in. If you are talking about a disc, then yes.
I guess if a sphere is defined as all the points with at certain radius from the center or closer, it would remain a sphere, but I'm not sure if that's the proper definition.
What do you mean? Are you talking about things like "great circles" ("orthodomes")?
It's not related as far as I can see? How do you figure?
Well, yes, obviously. But for that trick to work, you'd need to find an infinite circle in reality, and we know that currently can't be done (the observable universe being finite and all that).
Not being able to find a physical case of an infinite circle doesn't mean one can't be constructed in abstract, so no this part of your reasoning doesn't hold up.If not, it would not be possible, seems to me....
No, I just visualize a sphere as being an infinite collection of equal circumferences in all 3 dimensions, equi-distant from the center, completely enclosing the volume of the sphere?