Logical argument using infinity

[Speakpigeon]

There's an entire sub-field of mathematics devoted to transfinite numbers, so it's not "just" a concept. It can be made more precise. For example, the classic result is that the infinity of the real numbers is provably larger than the infinity of the integers - a result due to Cantor. i.e. not all infinities are the same, but some are.

There are different classes of infinite sets. You can compare these classes of infinities up to a point. That in itself doesn't make infinity a number, and precisely because we can conceive of different classes of infinity. Infinity? Which one? There is an infinity of ways of being infinite, and while we might be able to order any two types of infinity relative to each other, that still doesn't make infinities numbers. At best, we can think of them as an ordered set of types of infinity, with the possibility that two infinite sets be commensurable, like say N and Q, which arguably have the same size if not properly speaking the same number of elements.
EB

 

[arfa brane]

It's in the definition of "circle". For example, Google defines a circle as: "a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre)."

Then an infinite circle consists of points an infinite distance from a fixed point. But an infinite distance is not a fixed distance, is it? So the centre cannot be fixed (fixed by what?, I again ask)

If finite distances have no meaning, how can the width of your line have meaning?

Ah, well, what's needed there is an axiom of lines: a line is one-dimensional everywhere in the plane.

But let me illustrate. Let's go back to the center of the circle, and use this as the origin . . .

But you can't do that, the centre is the centre of the infinite plane . . .

 

[NotEinstein]

Then an infinite circle consists of points an infinite distance from a fixed point. But an infinite distance is not a fixed distance, is it?

What do you mean by "fixed distance"?

So the centre cannot be fixed (fixed by what?, I again ask)

To the best of my knowledge, fixed in this context means it's the same, unchanging point for all points of the circle.

Ah, well, what's needed there is an axiom of lines: a line is one-dimensional everywhere in the plane.

I'm pretty sure that's not an axiom, but a definition?

But you can't do that, the centre is the centre of the infinite plane . . .

Yes, I can do that. In fact, I did, in that very post. A circle has a center, by definition. Therefore, I can obviously start working from that center. If you are now arguing that infinite circles don't have centers, you are in fact saying infinite circles can't exist.

 

[arfa brane]

Therefore, I can obviously start working from that center. If you are now arguing that infinite circles don't have centers, you are in fact saying infinite circles can't exist.

You can start working from the centre of the plane? Ok.

Suppose I do that too. How do I find out where you started working, or that you started a finite distance from where I did?

Suppose I construct a nice straight line with zero curvature, and assert that I can extend it indefinitely at both ends. So far, so not at all controversial.

Now I assert that my line is an arc of a circle. Where is the centre of this circle? Where do I start working from, as you put it?
Do you tell me my line isn't a circle, and that's all there is to it? Can you prove this?

 

[NotEinstein]

You can start working from the centre of the plane? Ok.

Again again, what plane? I'm talking about the center of the circle.

Suppose I do that too. How do I find out where you started working, or that you started a finite distance from where I did?

Just locate the point that's equidistant from all the points of the circle, and you'll have found where I started working from. It's the same way I found that point.

 

[arfa brane]

And we seem to be going around in . . . circles here.

 

[NotEinstein]

And we seem to be going around in . . . circles here.

True; I made an argument several posts ago, and you still haven't come around to address it in any meaningful way. Instead, you've come up with your own argument why infinite circles can't exist.

Which is fine by me: if infinite circles can't exist, there's no doubt an infinite number of ways to demonstrate that. The conclusion remains the same.

 

[arfa brane]

This got pretty ludicrous a while ago (thanks everyone).

An infinite circle can't exist if it contradicts the definition of a circle as the locus of points equidistant from a given fixed point.
An infinite line however, can and does exist. Yes it does!
And according to a tenured professor at a university, so can an infinite circle (all it needs is an infinite plane).

Given the second statement, an infinite line can be defined as the locus of points an infinite (not fixed) distance from a centre (not fixed) of the plane.
So calling the line a circle is a convention, albeit one that seems to be contradictory. However, in the projective plane which is the plane plus the line at infinity such a "circle" is well-defined.

I'm not sure what I can do about that, probably nothing.
The Greeks started it, after all.

 

[arfa brane]

Again again, what plane? I'm talking about the center of the circle.

The plane the circle is embedded in?

Just locate the point that's equidistant from all the points of the circle, and you'll have found where I started working from.

Just locate a point which is an infinite distance from every point on the circle?

Please if you can, contradict my construction of a straight line which is part of an infinite circle. Then see if you can contradict the assertion that the straight line has a centre an infinite distance from every point on the line.

 

[NotEinstein]

The plane the circle is embedded in?Just locate a point which is an infinite distance from every point on the circle?

Ah OK, in that case, you are wrong. I did not start working from the center of that plane (if that center even exists); I started working (explicitly) from the center of the circle.

Just locate a point which is an infinite distance from every point on the circle?

Yeah; that should be easy for you, as you are the one claiming infinite circles exist, and this is literally just using its definition.​

Please if you can, contradict my construction of a straight line which is part of an infinite circle.

Before one can do that, first it needs to established that the concept of an infinite circle is even well-defined.

Then see if you can contradict the assertion that the straight line has a centre an infinite distance from every point on the line.

(No comment.)

 

[arfa brane]

Before one can do that, first it needs to established that the concept of an infinite circle is even well-defined.

Stand in the plane somewhere and assert you're in the centre of the plane (which if course there is no way to prove because the plane is infinite, therefore you can stand anywhere, and in fact that's where you are).
Now point in as many directions as you like and assert there is a circle of directions over the point you're on.

Now assert that this circle can be extended indefinitely, to an "horizon at infinity".

 

[NotEinstein]

Stand in the plane somewhere and assert you're in the centre of the plane. Now point in as many directions as you like and assert there is a circle of directions over the point you're on.

Now assert that this circle can be extended indefinitely, to an "horizon at infinity".

That's taking the limit to infinity, which, as has already been pointed out in this very thread, doesn't mean you'll ever reach infinity, or that what happens at infinity is properly approximated/predicted by that limit. Your argument is flawed.

 

[Speakpigeon]

That's taking the limit to infinity, which, as has already been pointed out in this very thread, doesn't mean you'll ever reach infinity, or that what happens at infinity is properly approximated/predicted by that limit. Your argument is flawed.

Yeah, and I think this shows an interesting distinction between constructive mathematics and traditional mathematics. You can do actual infinities in the latter, not in the former. I guess it's a case of trying to have your cake according to tradition and eat it constructively.
EB.

 

[NotEinstein]

Yeah, and I think this shows an interesting distinction between constructive mathematics and traditional mathematics. You can do actual infinities in the latter, not in the former. I guess it's a case of trying to have your cake according to tradition and eat it constructively.

I have to admit I'm not very familiar with the details about constructive versus traditional mathematics, but what you said sounds about right to me. It's not impossible to include infinity into mathematics (for example, the extended real numbers), but it does mean some of the traditional (including Euclidean?) results may no longer hold. Mixing the two leads to inconsistencies, contradictions, and bad conclusions.

 

[arfa brane]

That's taking the limit to infinity, which, as has already been pointed out in this very thread, doesn't mean you'll ever reach infinity, or that what happens at infinity is properly approximated/predicted by that limit. Your argument is flawed.

How is asserting I can extend a circle of directions indefinitely taking a limit? I just extend the circle; no limits involved and no calculus of differentiable curves. Your objection seems flawed.

I assert that the circle does reach infinity. What now? Can you prove it doesn't?
Of course I have to accept that constructing such a circle of directions isn't possible, unless the circle has a finite radius. But so what?

What "laws of mathematics" contradict my assertion that I can extend a circle of (well-defined) directions indefinitely, such that the distance to the circle is infinite?

 

[NotEinstein]

How is asserting I can extend a circle of directions indefinitely taking a limit? I just extend the circle;

In other words: you are taking the limit of the radius approaching infinity. That's what the procedure you are describing is.

no limits involved and no calculus of differentiable curves. Your objection seems flawed.

That you have missed the fact that you have accidentally taken a limit without calling it that, doesn't mean you haven't taken a limit. It's not my objection that's flawed; it's apparently your understanding of what limit taking is.

I assert that the circle does reach infinity. What now? Can you prove it doesn't?

You literally started with "I assert". Who do you think carries the burden of proof here?

 

[NotEinstein]

(Can you please stop adding to your posts after you've posted them? It's easy to miss stuff that way.)

Of course I have to accept that constructing such a circle of directions isn't possible, unless the circle has a finite radius. But so what?

If you admit you can't construct an infinite circle, then how can it exist?

What "laws of mathematics" contradict my assertion that I can extend a circle of (well-defined) directions indefinitely,

None, not a single one. In fact, I've explicitly mentioned this as being possible. This is the "taking the limit" approach.

such that the distance to the circle is infinite?

And that's the jump to infinity. Please provide evidence that this is allowed in this case.

 

[arfa brane]

In other words: you are taking the limit of the radius approaching infinity. That's what the procedure you are describing is.

No I'm not taking the limit, of anything. I'm extending the circle of directions is all.

That you have missed the fact that you have accidentally taken a limit without calling it that, doesn't mean you haven't taken a limit.

I haven't taken a limit. I haven't defined a function with a limiting value.

You literally started with "I assert". Who do you think carries the burden of proof here?

You asserted I "accidentally" have taken a limit. There is no function with a limit in my post.

 

[arfa brane]

If you admit you can't construct an infinite circle, then how can it exist?

How can the infinite plane exist? How do you construct it?

And that's the jump to infinity. Please provide evidence that this is allowed in this case.

The Euclidean plane is infinite. How did I make that jump? Is it to infinity?

 

[NotEinstein]

No I'm not taking the limit, of anything. I'm extending the circle of directions is all.

Almost right: you are extending the circle towards infinity. That's taking the limit of the radius to infinity.

I haven't taken a limit.

As I have now pointed out repeatedly, yes you have.

I haven't defined a function with a limiting value.

So the curvature of the arc doesn't approach zero, is that what you are claiming now?

You asserted I "accidentally" have taken a limit. There is no function with a limit in my post.

So the curvature of the arc isn't a function of the radius of the circle?