Actual Infinity in Reality?

[polymath257]

Well, and that is the whole point. There is NOT a start. It is always going on. Any time you 'set down' will always have an infinite amount of time in its past.

So, yes, the basic issue is that there is not a start. There isn't a contradiction there, just a mind bender.

That come back to how does one define time.   Here it seems to me that we are arguing from a hazy, notional perception of time, rather than from a more rigorous definition of time that could usefully define a direction of time.  It is necessary to first define what constitute direction of time before arguing whether time could stretch infinitely along that direction.

For time to be useful concept with specific direction, there must be some mechanism to mark the directional progress of time.   Time has no meaning if either nothing whatsoever changes, or if all changes are purely random such that defining time as going one way is utterly indistinguishable from defining time as going in the opposite direction.

All physical laws except one do indeed seem to work just as well going one way in time as going in the other.  So nothing else in our physical universe can be used to consistently mark the progress of time.   Anything the law says can happen going one way the same law says can go in the other.  

The only exception seems to be to be the law of entropy.  So that suggest the increase in entropy marks direction of time.   This makes it problematic to say time extends infinitely back into the past, because to trace time backwards is to trace reduction in entropy.   Entropy can not go negative, and is likely to be quantized.   So while one might notionally imaging time going back forever, beyond a certain point time no longer has meaning.   So one might say meaning time does not stretch back forever.

Some aspects of the weak force are time anti-symmetric, but they are unlikely to be relevant here.

The problem with entropy is that it is not fundamental. It is a statistical construct based on the loss of information in going from a low level (microscopic) description to a high level (macroscopic) description. This is the whole point of the various 'ensembles' of statistical mechanics.

There are two main physical arrows of time: entropy increase, and universal expansion. There seems to be no good reason for the two to be the same. So, in a pre-Big bang scenario, it is *possible* to have a contracting universe with increasing entropy. And some versions of quantum gravity have exactly that. We may also have long periods of expansion with zero entropy production (which satisfies your requirement for change to give a sense of time direction).

Another aspect of this is that most scenarios where time goes infinitely into the past are in some sort of multiverse model. In this, the entropy of the multiverse need not directly correlate to that of particular universes (there is the randomness between universes to consider). This allows for a very different entropy picture that the one we see locally.