Yes. I'm familiar with this brilliant argument. Time is infinite in the past because time is infinite in the past.
Nope. Time is infinite in the past because things do not appear out of nothing, so something has always existed. Always existed = existed for an infinite amount of time- an amount of time that cannot be measured.
It is saying time without end has ended at every present moment.
That is a pretty sloppy way of saying it, because obviously existence continues past every moment. In fact "moments" are simply a convenient, sloppy way of referring to parts of the eternal continuum. Moments are not truly separate from eternity.
Why is it so hard for some to understand that something that doesn't end can't have an end?
Of course eternal existence does not end. Even when we say that the end of a day is right before 00:00, this doesn't mean that time has ended. In eternity, and even reality, a day does not have a well defined beginning or end. We just set semi-vague boundaries...
I make no claims to know how time came about. I only know it is illogical to think it always existed.
You "know" something which isn't true. I don't see how "knowing" something that is untrue is useful for anything. I can see how one can create certain interesting ideas from a fictional idea, but to purport it as an actual truth muddies the water.
The idea "time has a beginning" should be taught in creative writing classes, and should not be given equal footing in a philosophical or scientific setting. Creation scientists, who have derived their ideas from a bible hold similar beliefs- you don't find modern cosmologists saying that there is an actual beginning to existence. Something always comes from something. The other side of the equation is that someone can create something out of nothing (but imagination, which is something).
In other words, the idea of nothing existing is created in the imagination. The idea of nothing is something that did not exist, yet was created as an imaginary concept to create the idea of a beginning, to create story lines, to be creative. There is a beginning of nothing- when someone first imagined what it would be to begin from a point at which nothing else existed. Of course nothing never actually existed, except as an imaginary idea.
If an infinite amount of time exists before any present moment in time then infinite time must have passed before any present moment in time. Time in the past is time that has passed. But infinite time never passes so it is impossible for there to have been infinite time before any present moment.
You can't define existence as having a start, because what started existence is not nothing. Nothing does not exist, it never did. It's impossible for it to exist.