Disclaimer. I am a mathematician, not a physicist. The extent of my physics training is a couple of high school courses. So, you may have to be very patient with me in understanding your answer(s)/comment(s).
Question. Was there a first moment in time? So, for instance, is the past best represented by an open interval $(t_0,t_p)$ (where $t_0$ denotes the would-be first moment in time, and $t_p$ denotes the present moment in time) or a half-open inteval $[t_0,t_p)$? Or something else?
Discussion. I am guessing that the answer to this question depends on what cosmological model is in play. The problem is, I don't know the first thing about which cosmological models there are to choose from, whether certain models are more likely to be "true" (for lack of a better word) than other models, etc. In fact, I only have a very rough idea of what a cosmological model is!
If the cosmological model of choice excludes a first moment in time, can it be extended easily to include one? (Maybe that question is ill-formed, in which case I apologize.)