What's half of forever?

[geokker]

As a finite sphere's surface is flat at the infinitesimal level, so is an infinite sphere's surface flat at the finite level.

This is simply incorrect.

[Guest loarevalo]

how? Please elaborate.

 

I enjoy being told I'm wrong - but I have to believe you. Give me reasons to believe you, don't just tell I'm wrong.

 

That's why I'm here in Hypography - to be told I'm wrong, to gain from else's insight, and possibly contribute to that too.

[geokker]

Sorry, I was pushed for time. If you travel a straight path on the surface of a sphere you must return to your origin. If it is truly infinite, this cannot happen. We are arguing absolutes here. Infinity is not just big, it's a different kettle of ball games altogether. Also, you must be able to calculate the centre of a sphere from a chord. If the surface is flat, you cannot plot a chord.

[Guest loarevalo]

So are you saying: Infinite spheres don't exist?

 

Whether they can exist objectively is irrelevant. Mathematically, infinite spheres are perfectly valid objects. That applied specifically to the number line or XY plane:

See Compactification:

"...For example, the real line is not compact. It is contained in the circle, which is obtained by adding a point at infinity. Similarly, the plane is compactified by adding one point at infinity, giving the sphere."

[geokker]

You're losing me. Are you saying you can have a sphere without curvature?

[EWright]

He is correct, you can have a flat local surface on a sphere, even one that is not infinite, but is really big. Also, parallel lines can and do intersect.

[Guest loarevalo]

You're losing me. Are you saying you can have a sphere without curvature?

I don't know about that. I'm guessing that a sphere must have curvature. An infinite sphere is curved - however, at the finite level, it's flat. A finite sphere is curved, but at the infinitesimal (local) level, the surface is flat. A triangle drawn on the surface of the Earth has a sum of internal angles > 180 deg. As such triangle gets smaller, the sum gets smaller approaching 180 deg at the limit - but at the local level, the sum equals 180 deg.

 

Those facts about a finite sphere can be applied also to an infinite sphere.

 

In regards to whether parallel lines intersect - it depends what geometry one uses. I would agree in that they do intersect. Euclid's fifth postulate (which states that paralel lines don't intersect) is independent from the other postulates, which means that paralel lines aren't necesarily disjoint.

 

Just recently in Analysis (Calculus), mathematicians are becoming more bold, and are coming up with theorems that explicitely make curves, flat the infinitesimal level. This is the principle of Microaffiness. This is very cutting edge, wikipedia has no entry on it - but you could look it up on google.

[geokker]

So if I'm standing on the surface of an infinite sphere, what space am I occupying? Presumably space into which the sphere can grow, becoming infiniter. ;)

[Guest loarevalo]

Ultimately the Universe, the collection of all thoughts, the One is self-contained, in the sense that while we can always increase, go to the next level, and then to the next, add infinitum, we cannot escape this Reality. Would it make sense to speak of what is outside of Existence? No, because there is nothing outside of Reality, because this "outside" doesn't exist. Everything is in Existence, inside this which we call "everything."

 

To put it mathematically, while we can always find a number greater than another, and then the next, add infinitum, without limit, we cannot get to a number that is outside this absolue scale. First of all, under this definition, there is no limit, there is no BIGGEST number - but one biggest number certainly exist only as the notion of the limit to where we cannot get. It makes no sense to speak of numbers beyond this limit, because by definition every (meaning EVERY) number is under this limit. Our absolute scale goes at the lower limit of "nothing" to the upper limit of "everything" - everything is in between by definition.

 

However, there are infinite numbers less than the "limit," so in mathematical space there could be infinite spheres that are "increasable" (unto a greater level of infinity) - these are not the types of spheres of which I am taking about.

 

We live on the surface of the earth, but there is a third dimension through which we can escape the Earth and leave its surface - doing this one would cease to exist on Earth (one wouldn't be on Earth). However, there is no extra dimension in Existence through which we can escape it, because then we would cease to exist. While everything exists or "stands" on the Universe, which could be conceived as a hyper-surface, we don't occupy space into that extra dimension thorough which we could escape; we are forever attached to the hyper-surface.

 

Your argument is that infinite spheres (or circles) cannot exist because they be must contained in a space, which would necesarily be bigger than the infinite spheres. Is flat space necesarily contained in another bigger space? Flat Eucledian XY plane of R+{-INF,+INF} is the same as an infinite square, which observations requires that we imagine a space containing said square (we can't think in absolute terms, only in relative terms). We just did the same to understand why we can't escape Existence.