Is there a cosmological model where the observable universe continues to shrink as we go back in time, and continues to expand as we go forwards in time, and such that at both directions it never reaches the mathematical limit of the change in that direction. So for example the lower limit could be time zero, which is never reached by the universe; i.e., as we go back in time, the universe continues to shrink and shrink infinitely, but each time it shrinks, it shrinks less than the prior time, such that $time_0$ is a mathematical limit to this shrinkage that is never reachable by the universe in the past, and similarly the upper limit might be some specific value $time_K$, in the future, where K is a finite real number, such that the universe expands towards $K$ but never reaches $K$, i.e. as time moves forwards (as we go more into the future) the universe does expand but the rate of expansion become lower and lower, but without stopping at all.

Mathematically speaking the above is consistent. For example:

$U = \{ 1/(2^n) | n=0,1,2,.. \} \cup \{ 2-(1/(2^n)) |n=1,2,..\}$

Here U is clearly an infinite set, yet its finitely bi-limited by 0 and 2.

Is there a serious problem with such a model?

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    $\begingroup$ You can write down any metric you want then calculate the stress energy tensor associated with it, but in general the stress energy tensor will correspond to a physically unrealistic form of matter. Offhand I don't know what the SE tensor would look like for the geometry you describe but it's going to be weird. $\endgroup$ – John Rennie Mar 20 at 9:12

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