# Logical argument using infinity

Discussion in 'General Philosophy' started by arfa brane, Mar 19, 2019.

1. ### SpeakpigeonValued Senior Member

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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

3. ### arfa branecall me arfValued Senior Member

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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)
Ah, well, what's needed there is an axiom of lines: a line is one-dimensional everywhere in the plane.
But you can't do that, the centre is the centre of the infinite plane . . .

5. ### NotEinsteinValued Senior Member

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What do you mean by "fixed distance"?

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

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

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.

7. ### arfa branecall me arfValued Senior Member

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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?

8. ### NotEinsteinValued Senior Member

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

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.

9. ### arfa branecall me arfValued Senior Member

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

10. ### NotEinsteinValued Senior Member

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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.

11. ### arfa branecall me arfValued Senior Member

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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.

12. ### arfa branecall me arfValued Senior Member

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The plane the circle is embedded in?
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.

13. ### NotEinsteinValued Senior Member

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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.

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

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

(No comment.)

14. ### arfa branecall me arfValued Senior Member

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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".

15. ### NotEinsteinValued Senior Member

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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.

16. ### SpeakpigeonValued Senior Member

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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.

17. ### NotEinsteinValued Senior Member

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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.

18. ### arfa branecall me arfValued Senior Member

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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?

19. ### NotEinsteinValued Senior Member

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

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.

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

20. ### NotEinsteinValued Senior Member

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

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

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

21. ### arfa branecall me arfValued Senior Member

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No I'm not taking the limit, of anything. I'm extending the circle of directions is all.
I haven't taken a limit. I haven't defined a function with a limiting value.
You asserted I "accidentally" have taken a limit. There is no function with a limit in my post.

22. ### arfa branecall me arfValued Senior Member

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How can the infinite plane exist? How do you construct it?
The Euclidean plane is infinite. How did I make that jump? Is it to infinity?

23. ### NotEinsteinValued Senior Member

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

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

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

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