frrostedman wrote:You're the expert so I will bow out on the point, but, technically speaking if a number increases in value and
never stops, that means it grows infinitely and that can equal nothing but infinity.
Ok I'll stop. You win. I begrudgingly hand you the trophy.
My agenda is not to win but to inform. I may be shot for saying this but your questions and challenges about this are better than I ever had the pleasure of witnessing while teaching calculus.
As for the point you raised, what if a variable increases in value but at an ever slowing and smaller rate? (I prefer to say a variable increases to a number increasing because it sounds like a number is changing when numbers don't change exactly.)
For example, this sequence of numbers is always increasing:
{1, 1.3, 1.33, 1.333, 1.3333, 1.33333, ....}
Yet all terms are bound by 1.4.
So that sequence does not diverge to infinity.
There's a theorem in calculus that says that if a sequence is bounded (in this case, it is bounded by 1.4)
and always increasing (ok, technically non-decreasing), then it must converge to some real number.
In this case, the "limit" of the sequence is what's known as the least upper bound of the sequence. 1.4 is an upper bound but not the least upper bound. 1.3333333.....(repeating) is the least upper bound for the provided sequence.
Just because I'm stubborn doesn't alone make me right. I always tried to encourage skepticism because it is when skepticism is addressed and satisfied (one way or the other) that genuine learning can occur. I think that's a general principle with skepticism, even in matters of faith. If the skepticism is not addressed then genuine learning will not occur, in my experience.
I think math is worthy of study for, among other things, the new avenues of thought that can develop. But when it is learned, I think one should be able to abandon mathematical thinking when one needs to.