The whole issue regarding the infinity of time is that the mentioned contradiction - the contradiction of the counted infinite number series - does not occur.
Time is not a physical thing itself. Physical reality only contains time meausurements, and all measurements of time involve only finite measures.
The construction of time without begin is done as follows. Firstly imagine an interval of time, of some unit length. Then we imagine that towards the begin of that time
interval, another time interval is attached. Now we have a longer time interval, but which also has an open begin. We repeat the step for that time interval,
and any subsequent time interval. The complete result is then a time line that ends in the "now" but of which any interval is connected on the side
of the begin with a previous interval, and so a beginningless time can be imagined. Whatever time interval we choose to consider, we always conclude
that towards the begin of the time interval, there is just another time interval connected, and also to that interval, etc. etc.
It means that there is no first time interval.
The objection that this time line without a beginning "can not be constructed ..."; but the objection argument bites itself in the tail, since it just can be stated in the addition of the constraint "... in finite time".
So, assuming time to be finite, time can not be finite. That is a paltry truth of course, but not an objection against an infinite time as such. Given no constraint on time,
why can time not be without a begin? There is nothing to object against that, and there is no physical or logical determination that can invalidate the idea.
We just assume that the universe - as it is in motion and change now - was in that state at all time, and that there is no way in which we can come from total motionlesness and changelesness into a state of motion and change, and assume therefore that there wasn't any specific moment in time in which there was no motion or change.
This is the most ordinary and simple thing to conclude, and to assume the opposite, one needs to explain sufficient reasons why one should assume that. There is no physical objection or logical objection against the idea that the universe was in motion and in change always, and since the measurement of time involves measuring change, this can neither object to the idea that time was without a begin, as the universe did not start out in total rest and motionelesness, but was in motion and change always.
