[sjastro]

Mirko,

I said.
We live on a (near) spherical surface.

Your response
We only know that because we can rise above it (in both theory and practice), but by doing that, we are introducing an external system of reference.

Our frame of reference is the same as the Earth's.
In this case it is in spherical coordinates defined by a polar and azimuthal angle which also locates our position on the surface.

I said.
We can travel around along a great arc on the Earth's surface and end up in the same spot we departed. Clearly this is not possible if the surface is infinite.

Your response.
But we could keep going indefinitely, couldn't we? We'd only because we, not physical constraints, have decided to "end" our journey there.

By nominating a departure point, and by further stating that we could only be there once, we, not "the Universe" have defined the limits.

The pathway is defined by the circle inscribed on the sphere. There are no limits. It doesn't matter how many times we travel around this circle, the radius of the circle remains the same. By calculating the radius through the circumferential distance travelled, we can find the surface area of the sphere which is finite.

I said.
Cosmologists have used a similar principle to test if the Universe in finite and spherical. A photon emitted could traverse a spherical Universe and end up near the same spot of emission.
Since the anisotropic regions of the cosmic radiation background can now be well resolved, double images in the background would indicate this phenomena.

Your response.
Just because this hasn't been observed doesn't mean it' not happening, since "well resolved" is clearly less than 100% resolved.

Edit: I might add that the further a photon travels, the less likely it is to maintain its original direction due to gravitational and other factors along the way. It would not seem unreasonable to think that given enough spacetime, a photon would eventually be turned back onto its origin, if not somehow rendered non existant in the process. As a result, perhaps the question of infinity is irrelevant because all possible trajectories of our photon are interlinked loops between which it can change. That way, it could avoid returning to the keen observer for a very long time but still be in a loop at any given time.

When I mean resolved, I'm referring to the angular resolution of the anisotropic detail in the cosmic radiation background.
There has been a progressive increase in detail from COBE to WMAP to the Planck probe.
Since the detail is now well resolved, cosmologists are able to measure the angular size of individual structures with greater precision.

The measurements are consistent with a flat univese.

Steven