a) if time has been around forever, there is a timespan with an unbounded past, unless you specifically define spans as finite quantities. Spans can be infinite, as far as I know.
Unless what you call a time span (in your single thread of existence) is actually multidimensional. I think GR can be interpreted in such a way that each point in spacetime has a unique timeline that splits off from the rest at every moment. So there is an infinite volume of time in every finite timespan, since an infinite amount of timelines are created each moment.
Or broadly speaking geometric, when we describe curves, or the number of faces of an infinihedron (you know, a sphere...), or number of sides of an infinigon (a circle).
In fact, it's directly built into smooth nature all around us, and there is nothing more natural than infinity, so it's really hard to avoid it when describing nature with mathematics. Pi, e, all things transcendental are required to transcend finite math so that math can be used to describe all of nature's infinities.
Without infinity, math cannot describe nature with anything resembling precision, because of the infinite amount of infinities we encounter in nature daily.
I'm all for infinite time spans. I would have more time to think about the infinite.
The infinite time I mentioned was more of a volume (lots of time lines) than a span. Sorry.
Now you'll have to explain to me how General Relativity could relevant to my question: Why the clock should show any particular reading at all, and if so, then which one?
It's not pertinent to that- you claimed that between any 2 points in time, a finite amount of time existed, drop the GR reference, and read it without invoking GR. It's not pertinent to that question still- it's pertinent to the claim that between 2 points in time, a finite amount of time exists, which is entirely dependent upon what you mean by "between 2 points in time".
If you say "the shortest time path between 2 points", there is only one. If there are multiple time paths of certain lengths between 2 points in time, or multiple shortest paths, then there can be an infinite amount of time between 2 points. I doubt there is- I'm just throwing out an annoying logical formality. I apologize.
With a clock that's supposed to have always been going on throughout an infinite past, there's no starting point, and it was never set to being with. And even if there was, without a finite time span between setting and reading, there's no good reason that the clock should read one particular value rather than any other.
Well, yeah. Depending on the configuration of all things in the universe, which might be somewhat location dependent, you're going to get different "first tick" times, even if the same type of clock (call it a photon) forms in various locations with various innate properties (say that a certain mathematical logic leads to certain conclusions, which lead to certain clocks being formed).
Before it forms, there might be various processes occurring that aren't measured by the clock. If conditions are right, and the clock can be formed, it is.
Obviously any clocks would not exist until someone knew that certain things evolved, propagated at relative rates, and figured out how to build something that used this fact to measure whether other things did as well, and then organized various stuff in the local universe. This requires some form of information persistence first, which requires an existing framework that can support information persistence, which requires also that the information that exists form specific "seed" values from which other information can crystalize into meaningful patterns, blah blah blah blah^infinity blah.
So the clock is meaningless. First, there must exist someone to measure, with the ability to measure, and influence the outcome of events, before there is a clock. Unless you propose spontaneous clock formation? I still remember when it rained watches in Ibiza, but that could have been pi "I begin to fall" casso's painting.... (casso.. latin... look it up, use circular reasoning).
Right now I don't think General Relativity has much relevance to that.
It doesn't, any framework in which time can occupy volume, instead of a "line", can have infinite amounts of linear time per volume. It's just GR has that sort of built into it (with spacetime expansion factored into it, which means even points separated by the a tiny finite distance experience different times because of the spacetime expansion between them...). Maybe it's the KK (Kaluza Klein) version that does, although I think they added dimensions of space, rather than time, which is sort of the same thing... if you think about it in a certain way.
Anyway. Drop GR from what I said, and just focus on "a volume has an infinite amount of parallel line segments within itself".