Some people don't want to say that there are just as many even natural numbers as natural numbers. But suppose that you and I will spend eternity singing numbers in harmony. You will sing every natural number in sequence: 1, 2, 3, ..., with a long pause for applause in between. And while you sing n, I will sing 2n. We will vary the speed of our singing to ensure that we take equal amounts of time. Clearly:
- The number of natural numbers = the number of your performances.
- The number of your performances = the number of my performances.
- The number of my performances = the number of even natural numbers.
- So, the number of natural numbers = the number of even natural numbers.
3 comments:
Couldn't we just take this sort of counter-intuitive result from mathematics as further evidence that actual infinities are impossible. After all, 2 things seem abundantly clear in this scenario:
1) We will never actually finish singing an infinite number of sounds. Indeed, we will never get any closer to actually completing an infinite number than when we started. So the whole scenario is based on something impossible.
2) If I were to match up the numbers you are singing with their counterparts in my list (your 2 with my 2, your 4 with my 4), and not speed anything up, then it would be evident that I'm singing more numbers than you are. And that would remain true until the moment we both transitioned from having sung a finite amount of numbers to having sung an infinite amount... which, again, we can never do.
So, why not think of this as just another reminder that infinites cannot exist, because, if they did, lots of absurdities would be true?
I'm not assuming the process will come to an end. I am talking of how many performances there will be. Surely it's possible that the future will be infinite.
You're talking of how many performances there will be by the time T=∞ which will never happen. The future may never have an end, but that doesn't mean its ever actually infinite. It's just bigger and bigger finites.
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