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Originally Posted by N1
That's exactly what I mean by rising above the surface. As soon as you do that, you see the curve. You have adopted an outside frame of reference. The point is to stay within the surface, a 2-dimensional universe. It's an analogy after all. Perhaps the problem of infinity can always be overcome by adding another dimension and looking at it from the "outside". You would need to step outside the universe though, as you've said to Alex below.
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Given that I am walking on the surface, not jumping or flying off it, nor drilling a hole through the Earth and travelling through it, my motion is purely confined to the surface.
There is no outside frame of reference in this example. I don't have to be outside the sphere to know its a sphere. I can measure its intrinsic curvature.
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The radius does not need to change for an infinitely long journey to be possible.
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That doesn't define whether a surface is finite or infinite.
A spherical surface is finite and unbounded. The surface of the Earth is finite irrespective how many times we travel around the circumference.
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Granted, this does not necessarily form part of the agument, but are we actually able to calculate the radius based on distance travelled, as opposed to obtaining an approximate value?
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Assuming the Earth is a true sphere we can measure the circumferential distance C travelled. Since the pathway takes on a great circle, the radius r is simply C/2*pi.
This also the radius of the Earth which has surface area of 4*pi*r^2 which is finite.
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That's the status as at today. I know of something else that was observed as being flat but later found to be (nearly) spherical 
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The difference here is the evidence itself rules out the Universe as a 3-sphere. Cosmology doesn't suggest the entire Universe has to be of a particular geometry to start with.
The options are flat, closed (spherical) or open (hyperbolic).
Steven