[jeffreyH (OP)]

Since a sphere has a bounding surface is it possible to have a sphere with infinite radius? If not then is it possible to have an unbounded space?

[chiralSPO]

Well...

There is no limit to how large a sphere can be (in theory). For any sphere of finite size, there are infinitely many possible spheres that are larger.

How you define a sphere then depends on whether there can be an infinitely sized sphere. For instance I think there wouldn't be any major problem defining some point as the origin of the sphere, and a locus of points all the same infinite distance away as the sphere (not very useful though). But I think that there would be many problems at the "surface" of a sphere with infinite radius. The local curvature would be related to the radius--and in the case of an infinite radius, I don't think there is any characteristic that would distinguish this case from a perfectly flat plane.

[wolfekeeper]

The Earths surface has no boundary, but is only 2 dimensional. So a hypersphere's surface would also be unbounded since it would have no boundary, while being 3 dimensional, but be a finite volume.

It's thought the universe might be aproximately hyperspherical.

[jeffreyH (OP)]

It's interesting that you equate the surface of the sphere with a flat plane. Would you consider this to be a plane that is infinite?

[wolfekeeper]

No, but it's unbounded.

[evan_au]

If the universe has a finite radius, then it can't contain a sphere of infinite radius.

However, if you include all points within the radius of the universe as "inside" the sphere, then there are no accessible points "outside" the sphere.

[petelamana]

Think about this:

Consider an n-sphere.  For any natural number, n, the n-sphere is defined to be embedded in a Euclidean (n+1)-sphere, where r ∈ +Reals.  This would then imply that a sphere of infinite radius is possible in (n+1)-dimensional Euclidean space, and might be defined as:

Sⁿ={x∈Rⁿ⁺¹:||x||=r}

There is one limiting factor here, though.  As the radius increases, the curvature of the sphere decreases.  As the curvature decreases it approaches the 0-curvature of a hyperplane.  However, considering the asymptotic nature of this restriction, as r → infinity, n-sphere → hyperplane should be an impossibility.

I think.   

[alancalverd]

A sphere is the locus of all points at radius r from the centre.

However large you make r, I can conceive of a sphere of r + Δr or even 2r, so you can't conceive of, let alone make, an infinite sphere.

[petelamana]

However large you make r, I can conceive of a sphere of r + Δr or even 2r, so you can't conceive of, let alone make, an infinite sphere.

Hi and Thank you for continuing the conversation.

I mean no disrespect, but the above statement seems to contradict itself.  If we create a sphere of radius r, then obviously any sphere of radius r+n would be possible.  Wouldn't a sphere of radius r+n, or just n, therefore be considered a sphere of infinite radius?  Help me out.  I admit I may be missing something.

[alancalverd]

No, because I can conceive of a radius r +n +1. Infinite means "nothing can be larger than....". Merely running  out of bricks doesn't mean you have built an infinite wall because, as the Panzer Divisions discovered, you can drive round it with a finite amount of fuel.

[puppypower]

If the radius of a sphere was infinite, then the diameter of that sphere would be twice infinite. Twice infinite contradicts convention, since infinite is defined as the last size.

There would not be enough room in an infinite universe, for a sphere of infinite radius. We could only fit about an eight of that sphere. The rest of the sphere would need to extend into other dimensions and would need seven other alternate infinite universes in a 2x2x2 configuration.

[petelamana]

An interesting argument is made by puppypower.  However, I must respectfully disagree.

A scaling factor, such as 2, would not change the limitless character of infinity.

2∞ = ∞ + ∞ = ∞

Nor would an infinite power.

2^∞ = ∞

[jeffreyH (OP)]

I'm curious to learn how to measure an eighth of infinity. @puppypower Care to elaborate?

[jeffreyH (OP)]

If the radius of a sphere was infinite, then the diameter of that sphere would be twice infinite. Twice infinite contradicts convention, since infinite is defined as the last size.

You have obviously never heard of positive and negative infinity. You need to brush up on your mathematics. Try studying limits.
https://en.m.wikipedia.org/wiki/Limit_(mathematics)

[puppypower]

I'm curious to learn how to measure an eighth of infinity. @puppypower Care to elaborate?

I used the concept of multi-dimensions and multi-universes, to extend infinity beyond the limits of just one infinite universe. The final shape, using eight universes; 2x2x2, will look like a single sphere of infinite diameter (not radius) that will be overlap all the eight universes. The affect of the infinite radius paradox, is felt by all; synchronized.

As an analogy, say I need to measure a broom handle that is several meters long, but in a single measurement. However, all I have is a 1 foot ruler. I may need to ask my friends to borrow their 1 foot rulers, so I can tape them all together until I get the length needed. Since all the rulers are needed to measure the broom, the length of the broom is a function of a multi-ruler collaboration.

This collaboration is not dependent on which ruler comes first, second or last. It only requires that they all are connected in a line. In the case of the universe, we end up with a type of holographic sphere of infinite diameter (not infinite radius)  common to all, but not at all points, since we each have a tangible eighth of a sphere and 7/8 that is not as easy to measure, yet there is impact.

[jeffreyH (OP)]

It is not sufficient to cherry pick scientific sounding words and string them together inside nonsense. It is much better to learn what the words mean. That way you can participate in intelligent debate. I would rather be having such debates with you than largely ignoring your contributions. It is not a weakness to be ignorant on a subject. It is an opportunity to learn. Learning can be very rewarding.

[puppypower]

If the radius of a sphere was infinite, then the diameter of that sphere would be twice infinite. Twice infinite contradicts convention, since infinite is defined as the last size.

You have obviously never heard of positive and negative infinity. You need to brush up on your mathematics. Try studying limits.
https://en.m.wikipedia.org/wiki/Limit_(mathematics)

Say we use negative and positive infinity. We start at the origin, and begin with an infinite line to define our radius. I will define this in the (1,1,1)  direction. This allows the radius to fit in an infinite universe. Since we are forming a sphere; locus of points equidistant from a single point, we need to move that radius in two angles, to cover the volume of the sphere. To do this, we will need to enter negative infinity territory on the other side of the origin. (-x,-y,-z)

If we fill this sphere in, we end up with as much negative and positive infinity. Relative to our diameter, since diameter in all directions is  r plus -r,  this equals zero. It cancels itself out.  It does not exist. To make it exist, we need to think outside a single universe box of negative and plus, to avoid plus and minus infinity cancelling. That is plus and  minus work in math. 

I have the mind of a contriver or engineer, so I am wired to figure out to make it work, instead of surrender to the limitations of traditions. I could not make it work, when negative cancels positive. This was considered. There was not enough room for a diameter of 2 times infinity in any positive only infinity quadrant. However, I could make it work using all positive, but that required only an 1/8 of the sphere, in each of 8 overlapping universes. The final shape is an infinite diameter universe hologram, in each of the 8 stations. The sacred nature of infinity is preserved. It is like the speed of light does not change.

[puppypower]

It is not sufficient to cherry pick scientific sounding words and string them together inside nonsense. It is much better to learn what the words mean. That way you can participate in intelligent debate. I would rather be having such debates with you than largely ignoring your contributions. It is not a weakness to be ignorant on a subject. It is an opportunity to learn. Learning can be very rewarding.

I agree with you. However, sometimes what is already there, cannot get the job done, without getting too confusing. This is when you may need to step outside the box. Outside the box is often harder to describe, since there is no tradition for a direct translation. The bridge often has to use common language, but with a slightly new definition added. 

[jeffreyH (OP)]

The number line can be thought of as ranging from minus infinity to positive infinity. You don't subtract the negative part from the positive part. It is a continuum and essentially unbounded.

[petelamana]

Hello, again...

There are so many ways to approach the definition of the infinite, yet those ways will all bring us to the same conclusion.  As a species we tend to want to quantify everything.  It is generally difficult for us to truly rely upon a purely conceptualized reality.  It is pointless.  We must either agree, or agree to disagree.  I am reminded of a book by Ralph Cudworth in the 17th century where he poses "how many angles can dance on the tip of needle."  Who cares?

Now, that is not to say that a debate about the scale and scope of infinity is pointless.  Infinity is a defined mathematical construct.  It is limitless.  Anyone who has an understanding of Lobachevskian geometry, limits, or even asymptotic functions has already accepted the boundless nature of infinity.  What is, in my opinion, pointless is attempting to quantify infinity.  By including any scaling factor in any discussion centered on infinity, such as 2∞, ¾∞, √∞, or even ∞^x, we are attempting the impossible.  Mathematics is, after all, a "religion."  In order for it to work certain assumptions, or axioms, must be accepted. 

For example:  1 + 1 = 2  Now prove it.

Wait a moment, before you throw me on the trash heap of the inane, allow me to clarify...

In order to prove 1 + 1 = 2 it must first be proven that more than one "1" can exist.  You cannot simply say "I have one orange here, and over there I have one orange, and that makes two "1"s.  The moment you identify having two seperate 1s you have "2".  So, it must be accepted that there can exist, at least, two distinct 1s.  Then you need to prove the concept of being able to "add" them together.  Not so simple an idea.  Why?  Allow me to continue...  If I had two distinct containers of water, and then I poured both of them into a single empty container, (adding them together), do I still have two distinct containers of water?  No.  I have something entirely different.  But if I had two distinct containers where one held water, and the other say - milk, and I "added" them in a third container, do I still have two distinct containers simply mixed together?  Yes.  It would be possible to "filter" the milk from the water.

My labored point is, and please forgive me for that labor, that we need to accept many things as being so, if we are to contemplate what could be.  I accept the concept of infinity.  I understand it.  It is boundless, limitless, dare I say "godly".  (little "g", not big "G".)  Once I accepted it, the fantastic scope of mathematics was unveiled.  Imagine where we would be today if Zeno of Elea had been able to accept the idea of a limit, 2100 years before Leibnitz?  Wow!!!