I'm a perpetual noob at this topic, so might be completely wrong, but: what I vaguely got from Hawking's books is that both black holes and the before-bang universe have matter in some kind of quark soup, in which our conceptions of space and time just don't exist. Since our spacetime depends on the non-soup layout of matter, the spacetime can be said to exist only with the universe after the bang (and until the crunch, if it comes). That's the age of the universe that we can reason about.

Curiously, Hawking still relates the hypotheses that (very roughly) calculate some kind of time that it would take for the soup to form into the proper universe—via Hilbert's spaces of configurations. I.e. these theories somehow count possible arrangements of matter in the soup, in the absence of space, and its movements from one arrangement to another, in the absence of time. That confuses me to this day.

You can still take limits as the Big Bang is approached, even if the limiting case of the Big Bang itself can never be reached. Just as you can take limits of things as some positive parameter approaches zero even if the parameter never actually takes the value zero.

Open Interval / Zeno's Paradox / White Holes / time passing more slowly closer to Big Bang.

I felt similarly. Thinking through the article's comparison between first moment in time and "smallest possible positive number", maybe it's sensible...

> even though there’s no first moment of time, we can still measure the age.

> even though there’s no first [positive number after zero], we can still measure the [range between zero and one].

That second variation doesn't seem so contradictory, so maybe the original isn't :)

The second variation does not seem equivalent to me. If there's no "first moment of time", you should be measuring the [range between negative infinity and one]. Using a fixed point like "zero" is the same as assuming there _is_ a first moment of time.