That’s not precisely what I meant and I don’t disagree with you on those technical Points. The Issue is the distinction between potential and actual infinity, and looking over what I wrote, it isn’t about countable versus uncountable. (Oops)
True in both your and John’s comments, it seems to me. Universals such as integers “exist” (how is a worthwhile discussion in itself). But the question is not whether rational numbers need to have a beginning, but whether a rational number of things (eg actual years, or seconds, or particles in the universe) could exist without a beginning.
When we were children we used to ask what the last number was, until we eventually twigged that it would always be possible to perfom “+1” to it. But if you have 100 fossils or stamps in your collection, saying “+1” makes no difference whatsoever.