Did physical reality (spacetime) begin to exist?

The BGV theorem is widely accepted.

• Alexander Vilenkin: “It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. With the proof now in place, cosmologists can no longer hide behind the possibility of a past-eternal universe. There is no escape: they have to face the problem of a cosmic beginning.” [Many Worlds in One (Hill and Wang, 2006), 176.]
• William Lane Craig & James Sinclair: “The Borde–Vilenkin–Guth (BVG) singularity theorem is now widely accepted within the physics community. As of this writing, it has gone largely unchallenged.” [The Blackwell Companion to Natural Theology (Blackwell, 2004), 142.]
• Gott-Li: “The question of first-cause has been a troubling one for cosmology. Often, this has been solved by postulating a universe that has existed forever in the past. Big Bang models supposed that the first-cause was a singularity… with a proper theory-of-everything, one could perhaps push through to earlier times. Inflation has solved some of these problems, but Borde and Vilenkin have shown that if the initial inflationary state is metastable, then it must have had a finite beginning also. Ultimately, the problem seems to be how to create something out of nothing.” [J. Richard Gott and Li-Xin Li, “Can the Universe Create Itself?” (1997), p.41]
• Audrey Mithani, Alexander Vilenkin: “Did the universe have a beginning? At this point, it seems that the answer to this question is probably yes. Here we have addressed three scenarios which seemed to offer a way to avoid a beginning, and have found that none of them can actually be eternal in the past. Both eternal inflation and cyclic universe scenarios have Hav > 0, which means that they must be past-geodesically incomplete. We have also examined a simple emergent universe model, and concluded that it cannot escape quantum collapse. Even considering more general emergent universe models, there do not seem to be any matter sources that admit solutions that are immune to collapse." [Did the universe have a beginning?, http://arxiv.org/pdf/1204.4658.pdf]

We know this because any eternally inflating space would require infinitely fast travel. This is relevant because an object traveling infinitely fast in space is impossible, producing absurdities and violating the laws of relativity.

• James Sinclair: "Vilenkin affirms that any universe (including universes modeled by higher dimensional cosmology, pre–Big Bang cosmology, and so forth,) which, on average, expands has to connect, in a finite time, to a past boundary (pers. comm. with Sinclair, March 4, 2004).
• Alexander Vilenkin: "Our argument shows that null and time-like geodesics are, in general, past-incomplete in inflationary models, whether or not energy conditions hold, provided only that the averaged expansion condition Hav > 0 holds along these past-directed geodesics. (Borde, Guth, & Vilenkin 2003, p. 3)39 A remarkable thing about this theorem is its sweeping generality. We made no assumptions about the material content of the universe. We did not even assume that gravity is described by Einstein’s equations. So, if Einstein’s gravity requires some modification, our conclusion will still hold. The only assumption that we made was that the expansion rate of the universe never gets below some nonzero value, no matter how small. This assumption should certainly be satisfied in the inflating false vacuum. The conclusion is that past-eternal inflation without a beginning is impossible. [Many Worlds in One (2006), 175)

The Borde-Guth-Vilenkin theorem proves any universe expanding on average, including Pre-Big Bang and higher dimensional cosmologies, has a beginning. It explains that if a universe could be expanding on average for a past-infinity of time, then an impossible object would need to be: an inertial object moving at infinity speed—faster than the speed of light.
Here is a simplification: Imagine a huge helium balloon with stationary ants scattered around it. Suddenly, one flying ant tells his friend, “I’m racing to the tip top to meet the queen!” His friend registers the flight speed as 4mph. However, the next ant he passes only registers his speed as 3 mph, and the next at 2 mph. What’s happening? The balloon is inflating! Similarly, in an inflating space, inertial objects (e.g. a bullet) are registered as slowing down relative to non-inertial objects that are carried by space (e.g. galaxies). From this we can prove all inertial objects have a finite past. If bullet-like are slowing down on average, then as you rewind the tape their speed is faster and faster. However, inertial objects can’t move infinitely fast. They can’t even move faster than the speed of light.
So, while inertial objects moving through inflating space for the entirety of the space’s existence is possible, it is impossible for an inertial object to have been moving through space for an infinity of time. Therefore, inflating spaces are not past-eternal.