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You are correct. Counting here changes its meaning. The correct form should be "all the numbers reachable by counting in finite time". E.g. you can count up t
by romwell 8y ago
You are correct. Counting here changes its meaning.
The correct form should be "all the numbers reachable by counting in finite time".
E.g. you can count up to 100 in a minute. Up to a million in a month[1]. Up to a billion in quite a long, but finite time. You take all such numbers, and you say that a number named Omega comes just after all of them (just like million and one comes just after all numbers that are less than or equal to ine million).
Omega is your first infinte ordinal - or, simply, infinity.
And now you're counting in a new way.
[1]https://www.mathsisfun.com/activity/count-billion.html https://www.mathsisfun.com/activity/count-billion.html
- navane 8y agoWhat I hear you say is that you can count to finite numbers in finite time, and infinite numbers in infinite time. That doesn't help me in understanding infinity.
- enedil 8y agoThe key point here is that you shouldn't be thinking of counting as a process that happens in time. How is infinity (here, omega) defined? We just say that it comes after any other natural number. So how to think about it? You shouldn't tell yourself: - ok, take number n, it's smaller than omega, so what about n+1, n+2, etc. ? You should be telling yourself - ok, take number n. It's smaller than omega. Now check any other number m. If this property holds for any given number, then omega is greater than any given number, and thus infinite.