Does the Borde-Guth-Vilenkin theorem definitively demonstrate that the Universe cannot be past-eternal, whatsoever? Does it not assume a classical space-time while the real world requires Quantum mechanics?
Are there successful models in Cosmology that are Past-Eternal?
Why might Alan Guth say the Universe might be eternal in the past, when he himself wrote a theorem in 2003 saying it most definitely isn't?
The simplest way around the usual interpretation of the BGV Theorem is described in Aguirre and Gratton's 2008 "Inflation without a beginning: a null boundary proposal", on the web: It requires dual arrows of time, pointing in opposite directions in de Sitter spacetime, which would both be of the usual thermodynamic variety. (I don't know if that might've been what Guth was alluding to, as I didn't see his remark and you didn't specify it, but Vilenkin, in a critique entitled "Arrows of time and the beginning of the universe" that's also available free on the web, found it plausible, even while claiming that it didn't invalidate the BGV theorem, which would apply equally to either half of the spacetime concerned.) De Sitter spacetime requires a contracting phase preceding the expanding one that's usually the concern of inflationary cosmologists, and, as Vilenkin points out, the backward or past-directed arrow would apply to the contracting one: Presumably the two together would have a net expansion of zero, which is another way it could stay in line with the BGV theorem, that only requires a beginning for universes with a net expansion greater than zero.
In his profoundly Christian blog, Aron Wall points out the fact that, since neurological processes are also thermodynamic, anyone in the backwards-pointing arrow might perceive time very much as we do.
A more recent proposal, involving dual arrows of time that are gravitational rather than thermodynamic, was worked out by Barbour, and is sketched in several magazine articles: Their descriptions of it can be found with his name and the term "Janus point", which is the center of time in his scheme.
Linde is well known to have disagreed with BGV, and it's not clear to me how it could apply to his "chaotic inflationary" universe, which nevertheless seems to involve such a range of scales in its regions that it is not renormalizable and cannot, in consequence, easily be compared with the false vacuum varieties of inflationary cosmology, in assessing its validity.
An even more recent and detailed cosmology that appears to provide for past eternality of a spatial and temporal multiverse is Nikodem J. Poplawski's "Cosmology with torsion", described, as of Mar.2019, in numerous papers (including a recent one published by the highly reputable journal "Physics Letters B"). As causal separations are, given the statistical increase in entropy (disorder) toward both the past and the future within the currently-observable portion of our region, the only factor that might leave a spatial-temporal multiverse that's eternal to the past with some predictability (i.e., without infinite entropy) would be permanent causal separations between regions each previously occupied entirely either by a relatively large star of the usual rotating variety, or by a sequence of them on decreasing scales of space and time: With temporary causal separations of the type last favored by Hawking, a beginning of time would be required.
Although Aguirre and Gratton's cosmology would accomplish such permanent causal separations as well as Poplawski's, I'd favor the latter's, because there are indirect signs that it's operative, particularly in the similarity between the observed surface of last scattering and the hypothesized inboard side of a black hole's gravitational horizon.
By representing what's been considered "the" big bang as the one on the local scale of an infinity of them, either Poplawski's theory or AG's would compare to the unmodified big bang theory as having a hypothetical explanatory power of doubled infinity, but Poplawski's would extend that increase to an asymptotically exponential extent, as the doubling factor would become an exponent.
As of July, 2019, the most recent cosmology that's eternal to the past is described (via "The European Journal of Physics") at https://link.springer.com/content/pdf/10.1140/epjc/s10052-019-6754-z. Formulated by Shabani and Ziaie, and making reference to earlier work by John Barrow and many others, it validates the basically static universe of Einstein's 1915 General Relativity by modifying that theory to incorporate some temporary oscillations (including inflation) through aspects of Brans-Dicke gravity, as well as through inclusion of the Einstein-Cartan theory's use of torsion (albeit in a more Friedmann-like setting than Poplawski's BH-based cosmology).
Attacks on past-eternality often rely on geodesics that are "incomplete to the past", so it's important to understand that Einstein-Cartan theory generally uses versions of "parallel transport" instead of geodesics: They're described a lot on PSE, recently at Geodesic equation in Einstein-Cartan manifold and its links (and their links)! Deviations from geodesics, in his cosmology, have more recently been discussed by Poplawski at https://arxiv.org/abs/1912.02173, and, in fact, linked to the Machian inspiration of General Relativity.
Although It's an idea sketchy enough that I've got to preface it with my factual "downvotes will cost you" disclaimer (which I learned the hard way), it's occurred to me today that, especially in a fully eternal and infinite multiverse, one or more cross-breedings of Poplawski's cosmology with Aguirre & Gratton's may have occurred through simultaneous gravitational collapses of the stars in one or more binary pairs: It occurs to me that such a situation might have unusually simple potential for explaining Dark Energy and/or Dark Matter, and I may need all the simplicity-mindedness I can get!
Nevertheless, at https://arxiv.org/pdf/1912.12926.pdf, I've found a Jan.20, 2020 paper by John Barrow effectively claiming that, since zero is finite and any finite action (including the start of the field-based inflation that's initiated by a bubble of repulsive gravity which Vilenkin has claimed arrives "from nothing") would be subject to restrictions listed in Barrow's "Conclusion", "past eternal inflating universes and future ever-expanding eternally inflating universes" are "ruled out", so that the spatial scale of any universe or multiverse that's eternal to the future would, presumably, have to be contracted, more-or-less like the sequential local universes of Poplawski's cosmos!
Poplawski's model appears to match a prescription by Luke A. Barnes (a postdoctoral researcher in astronomy and physics at the University of Western Australia, writing in Oxford's 2017 book titled "The Philosophy of Cosmology"), suggesting that "infinite multiverse modelers could try to manufacture a limiting process--perhaps a sequence of spacetime volumes", in attempts to resolve the so-called "Boltzmann Brains" problem: The sequential reductions in the scale of the "local universes" suggested by Poplawski's torsion-based cosmology might bring a quantum fluctuation in one iteration up to the size of that electromagnetic constellation which actually comprises a "mind" (or "brain") in a subsequent and scaled-down one.
A political style of problem that arises with past eternality is the fact that combinations of artifice and naturalism would be much likelier (if not certain) to have occurred in a past-eternal cosmology. For instance, the addition of mass to any large star nearing the time of its collapse to a neutron star might provoke its collapse to a black hole instead: As black holes radiate very little, such an addition of mass (which might, in extreme cases, comprise only a small amount) could be motivated by a desire to minimize exposure to unhealthy radiation on the part of colonists of a civilization not extraordinarily more advanced than our own, or it might involve a desire to provide for future replications of their own existence (at intervals perhaps random, and on scales probably smaller), by an entire civilization much further advanced, and prone to utlilize the possibility of such replications to impose a morality of "getting it right the first time" (and, consequently, "for all of future time"). Especially in the latter of these two cases, the separations between science and religion that dominate our civilization might be more difficult to maintain, and socially disruptive.