[MUSIC] Let's consider this limit. The lim (2x-x), as X → ∞ = ∞.
That's a true statement, but there's two totally different ways to think about
this statement, and it really hinges on the distinction between potential and
actual ∞. To have a potentially infinite pile of
fish is to have an endless supply of fish, as many fish as you'd like to have,
and that sense really goes will with how we're thinking about infinity in these
limits. To say the limit of 2x minus x equals
infinity, as x approaches infinity, is to say that I can make 2x minus x as big as
I like, as long as x. Big enough.
Contrast that with actual infinity. To have an actually infinite pile of fish
would be to have, right now, a pile of fish that contains infinitely many fish
at this very moment. Is it possible to combine that way of
thinking with infinity, with these kind of limit statements? Let me share with
you a fable, to see one of the paradoxes that results.
Once upon a time, there was a house. I lived in that house, with my cat.
And we lived, near a lake, and this lake is full of fish, so every day we went
fishing. And each day I caught two fish.
The first day I caught fish labelled 1 and 2.
Now, my cat prefers eating the lowest number of fish in our stockpile, so my
cat ate the fish numbered 1. The next day, I went fishing again, and I
caught fish labelled 3 and 4. And my pet, still preferring to eat the
lowest numbered fish. Eats fish number 2.
Another day another fishing expedition, I go fishing again, I get 2 more fish,
label 5 and 6, my cat preferring to eat the lowest number fish in our stock pile
eats fish number 3. I go fishing the next day and I get 2
more fish, fish label 7 and 8, my cat preferring to eat the lowest numbered
fish in our stock pile eats fish number 4.
And so it goes forever. Each day our stockpile gets bigger.
There's more fish in my pile every single day.
And yet, at the end of time, do any fish remain.
On the Nth day my cat ate the Nth fish. So which fish survives my cat's appetite?
Human beings want to understand infinity, but reasoning about actual infinity is
liable to walk us straight into those kinds of paradoxes.
Instead, limits by focusing our attention on potential infinity provide a way to
reason about infinity that avoid those kinds of paradoxes.
[MUSIC] It provides a way for mere human beings to think about infinity in a
precise way. [MUSIC]