This is a tough crowd, JesseM, with lots of moves. I would like to try one more time with pure logic - stated very simply so questions regarding semantics cannot cloud the issue.
The standard model assumes that the Big Bang occured ~13.7Gy ago, and it attributes three very basic qualities to the universe, that it is homogeneous and isotropic, and that there is NO privileged or special frame of reference in this universe. These are non-controversial aspects of the standard model, and I will confine the logical proof to these qualities.
Stipulation 1: We observe ourselves and our surroundings, including things beyond Earth. We are Observer "A".
Stipulation 2: Due to the finite speed of light, we see things as they were when light impinging our instruments left those objects. For instance, we see a star 10 Ly distant as it was 10 years ago. If it goes nova NOW, we will not know it for 10 more years. The most mature point in our observable universe is right here
, in the very center of our observable universe, 13.7Gy from the surface of last scattering as it appears to us
Stipulation 3: Judging from their redshifts, we see some distant objects as they were 13Gy ago, less than a billion years after the surface of last scattering.
Now for the logical proof:
Choose a quasar or galaxy at an apparent distance of 13Gly. Given the concordance assumptions of homogeneity, isotropy, and no special frame of reference, what can we say with certainty about a theoretical observer "B" who exists at that distant position right NOW?
We can say:
1. Since the universe is homogeneous and isotropic, and because "B's" frame of reference is no more or less special than ours, our theoretical observer looks out at his universe and sees a universe that is identical in its basic qualities to the one we see. He sees his own neighborhood, and due to the finite speed of light, he sees distant objects as they appeared in the past. Like us, he can only see objects out to about 13 billion light years distant. Anything much further, and he is looking at his
surface of last scattering, just like we look out at our own. Just like us, "B" has a visible universe about 27 billion light years in diameter. We are on one edge of his visible universe, just as he is on one edge of our visible universe.
2. Over half of the volume of our visible universe (a ~27Gly diameter sphere) is outside the observer "B's" visible universe and is invisible to him. Over half the volume of "B's" visible universe is outside our visible universe and cannot be detected by us. It may help to imagine these visible universes as a pair of interconnected spheres that overlap one another just a bit more than one radius (~13.7Gy)
3. If observer "B" looks in the direction opposite
that of our galaxy, he will be able to see other galaxies ~13Gly distant, and a hypothetical observer "C" in one of those galaxies will be able to look out and see a universe that is identical in its basic properties to the universes that observers "A" and "B" see. This is guaranteed by the three basic properties of the BB universe assumed in the introduction. Except for a very tiny intersecting volume centered on the location of observer "B", no part of the visible universe of observer "C" is in our visible universe (we are at observer position "A"), and except for same that tiny (lenticular, obviously) slice of space, observer "C" can see no part of our visible universe.
4. In a BB universe that is homogeneous, isotropic, and devoid of preferred reference frames, this logical iteration can be carried out forever, projecting to an infinite number of "visible universes" each centered on a unique observer. Therefore, if the BB universe is flat or open (and most adherents of standard cosmology are solidly wedded to flat at a minimum, and perhaps open), it must also be spacially infinite.
This is a logical proof derived from the principles of the standard model. I would attempt to simplify it further, but refrain for fear of loss of coherence.
How could the BB universe possibly be finite? To model a finite
BB universe, either at least one the three assumptions made by the standard model about the basic qualities of the universe must be wrong, OR the universe must assume a complex topology that somehow both keeps the universe flat/Euclidean locally AND bends space in such a way that one can set off in one direction and come back upon one's previous location without deviating from a straight path. Such theoretical topologies are apparently not falsifiable by any means, and absent any compelling reason to embrace them (apart from sheer revulsion at the thought of infinities
) there is presently no need to regard them as anything more than mathematical curiosities.
I welcome any logical refutation of this proof. "Carpet-bombing" this post with citations that do not address the logic of the proof and simple nay-saying will be cheerfully ignored.
Is there a logical failure in this proof? I would love to see it.