What is the difference between an infinite set of 1 dollar bills and an infinite set of 20 dollar bills?

Question

Even though both sets approach infinity at different increments, do they eventually approach infinity at the same value at the same degree? Or is the second set of infinite 20 dollar bills 20 times larger than the infinite 1 dollar bills?

Answer 1

Assuming that both sets are countable,they have the same cardinality and they both are in one-to-one correspondence with the set of natural numbers. The truth of matter is that they will never reach infinity in finite time.

Answer 2

Certainly, $f(n)=n$ (the amount of money in $n$ bills each worth $\$1$) and $g(n)=20n$ are both functions that "tend to infinity" in the sense that if you want the output to be greater than some large number $M$ than you just have to require that the input is larger than some number $N$.

I think the answer(s) to your question are a matter of perspective.

• Since they both tend to infinity, they're kind of the same in that way. Relatedly, in a Calculus class you might write that the sum of the values of $f$ and $g$ are both "$\infty$".
• But for $n\ge1$, $g(n)/f(n)=20$, so $g(n)$ could be said to grow ("approach infinity"?) twenty times as fast.
• However, they both grow linearly. Using "Big Theta", we could write $f(n)=\Theta(g(n))$ to express the idea that they grow at roughly the same speed.