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We assign the label ℵ₀ to an infinite set when we want to express the property that any given element can eventually be reached in a finite time if we keep coun

We assign the label ℵ₀ to an infinite set when we want to express the property that any given element can eventually be reached in a finite time if we keep counting for long enough. That's what it means, no more and no less. I don't see what it has to do with "assigning a label to infinity" or "assuming you can finish counting them". We don't use cardinalities to magically make the infinite finite (well, I don't, but I'm sure some pop-math explanations do), we use them to compare and contrast the properties of infinite sets, and refusing to acknowledge these properties doesn't make them go away.

nico 4y ago
> We assign the label ℵ₀ to an infinite set when we want to express the property that any given element can eventually be reached in a finite time if we keep counting for long enough. Look at your definition: “if we keep counting long enough” You are reducing infinity to “long enough” and using that as a label for infinity. You can’t know ahead of time that you will be able to finish without finishing. You are saying you can know the unknown. But you won’t until you go and try to count the numbers. You will never be able to count all the natural numbers. Regardless of what you think the formulas/symbols mean. Math doesn’t happen by itself, it needs an interpreter to “execute the operations”. Without an operator, math is just a bunch of symbols. The operator will never finish counting natural numbers. Or any other sequence/set/group of numbers that is infinite.
- LegionMammal978 4y ago
  Perhaps I have not been as clear as I should be. The sequence of counted numbers is infinite, and we would never be able to finish counting all of it. But all we care about for defining ℵ₀ is whether or not the sequence contains some particular value. If it does contain the value, then we can confirm that in finite time, even though the sequence is infinite. Let me try reframing it a bit. First, you think of some natural number N (e.g., 978) and keep it to yourself. Then, I start counting, "0, 1, 2, 3, 4, 5, ..." and never stop. You listen to me count until you hear N. Then, you can stop listening, even though I'll never stop counting. At that point, you already know that my sequence contains N, and you don't have to keep listening. I hope you agree that no matter which N you choose, you'll hear me say it sooner or later, as long as I keep counting up by 1. If N is a smaller number, you'll hear me say it sooner; if N is a bigger number, you'll hear me say it later. But there's no N you can choose that forces you to listen to me forever. (And again, just because you stop listening doesn't keep me from talking forever.) The same thing cannot be done with the real numbers. If you think of some real number R that I don't know, then there's no way for me to count through the real numbers to guarantee that you will eventually hear R and stop listening. Even if I do try to pick a certain sequence, there's a very good chance that you'll never hear R and you'll have to keep listening forever.

nico 4y ago

> We assign the label ℵ₀ to an infinite set when we want to express the property that any given element can eventually be reached in a finite time if we keep counting for long enough. Look at your definition: “if we keep counting long enough” You are reducing infinity to “long enough” and using that as a label for infinity. You can’t know ahead of time that you will be able to finish without finishing. You are saying you can know the unknown. But you won’t until you go and try to count the numbers. You will never be able to count all the natural numbers. Regardless of what you think the formulas/symbols mean. Math doesn’t happen by itself, it needs an interpreter to “execute the operations”. Without an operator, math is just a bunch of symbols. The operator will never finish counting natural numbers. Or any other sequence/set/group of numbers that is infinite.

LegionMammal978 4y ago
Perhaps I have not been as clear as I should be. The sequence of counted numbers is infinite, and we would never be able to finish counting all of it. But all we care about for defining ℵ₀ is whether or not the sequence contains some particular value. If it does contain the value, then we can confirm that in finite time, even though the sequence is infinite. Let me try reframing it a bit. First, you think of some natural number N (e.g., 978) and keep it to yourself. Then, I start counting, "0, 1, 2, 3, 4, 5, ..." and never stop. You listen to me count until you hear N. Then, you can stop listening, even though I'll never stop counting. At that point, you already know that my sequence contains N, and you don't have to keep listening. I hope you agree that no matter which N you choose, you'll hear me say it sooner or later, as long as I keep counting up by 1. If N is a smaller number, you'll hear me say it sooner; if N is a bigger number, you'll hear me say it later. But there's no N you can choose that forces you to listen to me forever. (And again, just because you stop listening doesn't keep me from talking forever.) The same thing cannot be done with the real numbers. If you think of some real number R that I don't know, then there's no way for me to count through the real numbers to guarantee that you will eventually hear R and stop listening. Even if I do try to pick a certain sequence, there's a very good chance that you'll never hear R and you'll have to keep listening forever.