scratching the surface of an infinite earth.

Roundy reminded me of something i was thinking about a long time ago.

infinite earth theory is often thought its an infinite plain. however does it need be? you have to hit this with an open mind, its more philosophy, but thats science anyway.

what if the earth is more akin to a klein bottle? or möbius strip? or real projective plane?

sounds crazy? yeah probably, but using such examples can explain a lot of strange things. for example;

Q what happens when a person travels over the edge?

A from the observer nothing has happened. they appear on the other side as if the world was a sphere. but due to klein theory (im coining that for an earth shape) the earth remains flat yet can be navigated like a sphere.

im just wondering if others have thought about this and have more advanced theories on it. if anything an answer like that would sure ruffle some feathers!

I had to read up what a Klein bottle is. If I understood that article right, your theory implys that earth is a non-orientable surface embedded in a 4-dimensional manifold.

Ingenious.

Why is a 4th spacial dimension necessary? I'm supposing that its because we would notice if we were on such an irregular shape if it were merely 3-dimensional. As long as the Earth is large enough as so the shape is subtle enough to avoid physical anomalies that would give it away, I don't see why this can't happen in 3 dimensions.

When you construct a Klein bottle in a 3-dimensional Euclidian manifold its surface intersects itself somewhere. If that astonishing phanomenon would occur on the surface of earth, tourist agencies would have noticed.

Read the book called "Worlds Beyond the Poles" by Amadeo Giannini. The infinite plain is very simple and therefore makes more sense, but in any case the physical continuity of the universe is much more appealing then an isolated sphere in dark space.

I'm not too convinced an infinite plane can exist. An infinite plane, by definition, would have an infinite number of points. So let I = the set of all discrete points on an infinite plane, {(0,0)...(∞,∞)}. Hence, the cardinality of I = |I| = ∞. But sets with infinite cardinality can't really exist in reality. Take a variation of Hilbert's hotel paradox: if we move every element n to n + 1 and add a new element to the beginning, then we've added one more element, which increases the cardinality by 1. However, the cardinality remains the same, because every element can be replaced. Thus, we have a contradiction in an infinite set.

I'm not too convinced an infinite plane can exist. An infinite plane, by definition, would have an infinite number of points.

How many points are on a finite plane?

Quote from: Arctangent on April 12, 2013, 07:22:16 PM

I'm not too convinced an infinite plane can exist. An infinite plane, by definition, would have an infinite number of points.

How many points are on a finite plane?

You're confusing a potential infinity with an actual infinity.

You're confusing a potential infinity with an actual infinity.

You said:

An infinite plane, by definition, would have an infinite number of points.

I said:

How many points are on a finite plane?

You didn't respond to my question.

Quote from: Arctangent on April 12, 2013, 08:07:54 PM

You're confusing a potential infinity with an actual infinity.

You said:

Quote

An infinite plane, by definition, would have an infinite number of points.

I said:

Quote

How many points are on a finite plane?

You didn't respond to my question.

I understood the point you were trying to make. The answer is, of course, a potentially infinite amount of points. Hilbert's famous paradox concerns an actually infinite set, however, which is what an infinite plane would instantiate as well.

So you say a finite plane contains a potentially infinite number of points, whereas an infinite plane would contain an actually infinte number of points. Right?

So you say a finite plane contains a potentially infinite number of points, whereas an infinite plane would contain an actually infinte number of points. Right?

That's pretty much just set theory. The contention of whether infinite sets can exist in reality is what's debated in philosophy of mathematics.

Quote from: Homesick Martian on April 12, 2013, 09:17:15 PM

So you say a finite plane contains a potentially infinite number of points, whereas an infinite plane would contain an actually infinte number of points. Right?

That's pretty much just set theory. The contention of whether infinite sets can exist in reality is what's debated in philosophy of mathematics.

Hilbert's Hotel contains an infinite number of rooms.

A (finite) plane contains an infinite number of points.

Why is the former infinity actual, but the latter potential?

Quote from: Arctangent on April 12, 2013, 09:39:49 PM

Quote from: Homesick Martian on April 12, 2013, 09:17:15 PM

So you say a finite plane contains a potentially infinite number of points, whereas an infinite plane would contain an actually infinte number of points. Right?

That's pretty much just set theory. The contention of whether infinite sets can exist in reality is what's debated in philosophy of mathematics.

Hilbert's Hotel contains an infinite number of rooms.

A (finite) plane contains an infinite number of points.

Why is the former infinity actual, but the latter potential?

Hilbert's Hotel contains an actually infinite number of rooms. A finite plane contains a potentially infinite number of points. The former infinity is actual because it contains an infinite number of discrete elements; the latter is potential because it contains a finite number of discrete elements, with a continuous infinity in between each discrete element.

For instance, take the distance between your eyes and the computer screen. The number of points approaches infinity, but the distance is still finite. So the finite distance "contains" (pardon the weak terminology) the infinity. Now consider an infinite distance from your eyes. This is an actual infinity, because it instantiates the infinite set with an infinite number of discrete elements, not merely approaching it.

This is the distinction made in philosophy of mathematics.

On a side-note, why are you arguing with me on this?

On a side-note, why are you arguing with me on this?

You came up with that, adressing the problem, if earth can be an infinite plane, or if it is philosophically (mathematically?) impossible. I just question your terminology.

For example: a plane

contains a finite number of discrete elements, with a continuous infinity in between each discrete element.

I never heard that a plane contains a discrete number of elements. With some infinity between them. What are these discrete elements?

Quote from: Arctangent on April 12, 2013, 09:59:52 PM

On a side-note, why are you arguing with me on this?

You came up with that, adressing the problem, if earth can be an infinite plane, or if it is philosophically (mathematically?) impossible. I just question your terminology.

Perhaps I should have initially modified "infinity" with "actual" so as to address the distinction before you brought it up. There are philosophers of mathematics on both sides of the issue, who believe an actual infinity can't exist (for example, Gauss believed that we only approach infinity, this is the view I hold; more recently, the philosopher of religion William Lane Craig has argued for the finitude of the Universe based on Hilbert's paradox) and there are those who believe it can exist, often for theistic reasons, to apply it to the Divine.

For example: a plane

Quote

contains a finite number of discrete elements, with a continuous infinity in between each discrete element.

I never heard that a plane contains a discrete number of elements. With some infinity between them. What are these discrete elements?

This time, I'll attempt to explain using interval notation. Say we have a distance of 3 units. We'd write this mathematically as [0,3]. In this way, we have a distance with a potential infinity -- the number of points in this distance approaches infinity, but it is in a sense "contained" within that 3 miles. This is the reason we can traverse finite distances -- we aren't actually traversing an infinit e number of points.

Now take an infinite distance. We'd write this mathematically as [0,∞). In this way, an actual infinity is instantiated. It isn't merely approaching infinity -- there are actually an infinite amount of units in this distance. It may be hard to grasp, but think of it conceptually, in terms of distances.

Have you ever heard of the paradox of Achilles and the Turtle? Aristotle's solution to the paradox was to distinguish between these two infinities -- one that is finite, merely approaching an infinity within the certain distance (i.e. [0,3]) and one that is actually infinite, instantiating the infinity as its distance (i.e. [0,∞)). It's an important distinction to make. Hilbert's Hotel is a paradox that applies to the latter, because it's not a finite "distance," it's an infinite "distance." Perhaps I was throwing you off with the "discrete" and "set" terminology.

Ok, that makes sense.

Also, I found this (in case somebody follows this thread):

http://www.iep.utm.edu/zeno-par/#H4

Ok, that makes sense.

Also, I found this (in case somebody follows this thread):

http://www.iep.utm.edu/zeno-par/#H4

LOL, yeah, the IEP can probably explain it better than I. Achilles and the Tortoise perplexed a lot of people, so Aristotle's duality of potentials and actuals (which extended beyond his treatment of mathematics, and into metaphysics) is highly important.

So yeah, an infinite plane as some of the Flat Earthers on this thread support leads to contradictions due to paradoxes in basic set theory. I don't support it.

I don't think that we can a priori decide if the universe (or the earth plane in this case) is infinite or not. But it is interesting to figure out what it would imply for the nature of reality, if it is.

I don't think that we can a priori decide if the universe (or the earth plane in this case) is infinite or not. But it is interesting to figure out what it would imply for the nature of reality, if it is.

Well, we know the Law of Non-Contradiction a priori. Hence, we know that contradictory properties like "married bachelor-ness" or "square circularity" don't exist. We can reject the existence of these phenomena a priori. If you don't see that, then there's really no way I can explain it. The Law of Non-Contradiction is just something we know; it's intrinsically justified. This is the reason I reject the infinite plane a priori.

Let's just say that I'm missing something, and that an infinite plane is possible. We'd never actually know that the infinite plane is possible a posteriori, either, since we'd never be able to reach infinity and empirically confirm it.

Quote from: Homesick Martian on April 12, 2013, 11:20:17 PM

I don't think that we can a priori decide if the universe (or the earth plane in this case) is infinite or not. But it is interesting to figure out what it would imply for the nature of reality, if it is.

Well, we know the Law of Non-Contradiction a priori. Hence, we know that contradictory properties like "married bachelor-ness" or "square circularity" don't exist. We can reject the existence of these phenomena a priori. If you don't see that, then there's really no way I can explain it. The Law of Non-Contradiction is just something we know; it's intrinsically justified. This is the reason I reject the infinite plane a priori.

An actually existing infinity is not contradictory in the same sense as a square circle.
"A square is not a circle" is an analytical proposition a priori, whereas "an actual infinity cannot exist" is a synthetical proposition a priori. So in fact you confuse formal and transcendental logic.

Quote from: Arctangent on April 12, 2013, 11:36:36 PM

Quote from: Homesick Martian on April 12, 2013, 11:20:17 PM

I don't think that we can a priori decide if the universe (or the earth plane in this case) is infinite or not. But it is interesting to figure out what it would imply for the nature of reality, if it is.

Well, we know the Law of Non-Contradiction a priori. Hence, we know that contradictory properties like "married bachelor-ness" or "square circularity" don't exist. We can reject the existence of these phenomena a priori. If you don't see that, then there's really no way I can explain it. The Law of Non-Contradiction is just something we know; it's intrinsically justified. This is the reason I reject the infinite plane a priori.

An actually existing infinity is not contradictory in the same sense as a square circle.
"A square is not a circle" is an analytical proposition a priori, whereas "an actual infinity cannot exist" is a synthetical proposition a priori. So in fact you confuse formal and transcendental logic.

No, I believe you're confusing multiple things here. I'm arguing that the cardinality of an infinite set is self-contradictory in the same sense that the shape of a square circle is self-contradictory. I'm not confusing synthetic and analytical propositions at all, because you're removing the rationale for my assertion.

 I'm arguing that the cardinality of an infinite set is self-contradictory in the same sense that the shape of a square circle is self-contradictory.

The cardinality of an infinite set is not self-cotradictory at all. Infinite sets are mathematical concepts and you can deal with them. There is no contradiction formally. What you mean is, that infinite sets cannot actually exist in the real world. And its here where your judgement is synthetical, not analytical. But you cannot make synthetical propositions a priori about the real world, so you cannot a priori prove, that the earth can't be infinite. That much I learned from Kant.

Quote from: Arctangent on April 13, 2013, 12:14:01 AM

 I'm arguing that the cardinality of an infinite set is self-contradictory in the same sense that the shape of a square circle is self-contradictory.

The cardinality of an infinite set is not self-cotradictory at all. Infinite sets are mathematical concepts and you can deal with them. There is no contradiction formally. What you mean is, that infinite sets cannot actually exist in the real world. And its here where your judgement is synthetical, not analytical. But you cannot make synthetic propositions a priori about the real world, so you cannot a priori prove, that the earth can't be infinite. That much I learned from Kant.

The cardinality of infinite sets are self-contradictory, because n =/= n + 1. This is exactly what Hilbert was trying to explain, and it's the reason what's called "naive" set theory was reformulated in the early twentieth-century.

Quote from: Homesick Martian on April 13, 2013, 12:33:01 AM

Quote from: Arctangent on April 13, 2013, 12:14:01 AM

 I'm arguing that the cardinality of an infinite set is self-contradictory in the same sense that the shape of a square circle is self-contradictory.

The cardinality of an infinite set is not self-cotradictory at all. Infinite sets are mathematical concepts and you can deal with them. There is no contradiction formally. What you mean is, that infinite sets cannot actually exist in the real world. And its here where your judgement is synthetical, not analytical. But you cannot make synthetic propositions a priori about the real world, so you cannot a priori prove, that the earth can't be infinite. That much I learned from Kant.

The cardinality of infinite sets are self-contradictory, because n =/= n + 1. This is exactly what Hilbert was trying to explain, and it's the reason what's called "naive" set theory was reformulated in the early twentieth-century.

By the way, that self-contradictory phenomena cannot exist is an a priori analytic proposition; but applying it to the self-contradictory nature of the cardinality of an infinite set is an a priori synthetic proposition. Not all a priori propositions are analytic.

Not all a priori propositions are analytic.

Did I say that?

I see no formal contradiction in infinite set theory and Hilbert didn't, too. Now you confuse contradictory with counter-intuitive.

Quote from: Arctangent on April 13, 2013, 12:40:22 AM

Not all a priori propositions are analytic.

Did I say that?

I see no formal contradiction in infinite set theory and Hilbert didn't, too. Now you confuse contradictory with counter-intuitive.

I see the contradiction; while Hilbert's Hotel doesn't itself justify the contradiction, there's other reasoning that can be employed in this case, as well. I'm off to sleep, I'll let you have the last word (I'm not going to respond and I grow tired of this thread to begin with, haha).

Fair enough.

Well, I find it naive to think, one can proove just by plain thought things like "can the universe be infinite". That's all I wanted to say.

Arc tangent likes to do that.