There is no largest or smallest real number. Therefore a set without a largest or smallest member is not necessarily empty.
Alternatively, being the smallest in the set of uninteresting numbers does not make a number interesting, at best it is unique like every other number and therefore uninteresting.
Yes. After the heat death, nothing will ever move or change and so it the universe will remain in a frozen state. Time will have stopped.
Sine time is relative based on the observer and observed, if neither change in any way (no “work” happened and no information is exchanged, even on the level of fundamental particles (or fields/loops/strings or whatever happens to be at the bottom)), for every possible combination of observer and observer, then time has not advanced in any meaningful sense.
The interesting number paradox applies to natural numbers, not real numbers.
> Alternatively, being the smallest in the set of uninteresting numbers does not make a number interesting
You are taking this too seriously. :-) Everyone would agree that "interesting" is subjective. This is supposed to be a humorous paradox, not a mathematical fact!
It is obvious that dates are not natural numbers. What is your point?
A set of dates obviously has an earliest date. The humour here is that an earliest date within the set of uninteresting dates is itself an interesting date by virtue of being the earliest such date in that set.
Uhh, no you’re thinking in terms of a finite set. From the set of all dates, what’s the latest possible date, or conversely the earliest possible date. The temptation is to say infinite BC is the earliest possible date, but infinity is not a number.
Infinity is not a number, it's a set of numbers, but specific infinities (e.g., the cardinality of the natural numbers, aleph-naught) are individual numbers, just not finite numbers.
But there is no infinite BC, right? There cannot be any date earlier than the date of the big bang. That establishes a lower bound for the set of dates, does it not?
The Big Bang is an apparent discontinuity, but we have no idea if a universe existed before this one. Alternatively, multiple universes could exist and those alternate universes could be much older. So at least theoretically things much older than our universe could exist.
Alternatively, being the smallest in the set of uninteresting numbers does not make a number interesting, at best it is unique like every other number and therefore uninteresting.