> (or insert them in order),
There is no order. You can't describe one, and it is proven that no order exists.
> you can count as many real numbers as you want
No, you can't count more than 0% of them, even in infinite steps.
The same way, I can just generate random real numbers, then define my set as in ascending order, and insert them in that order.
Which is what we essentially do with natural numbers as well. Except that the order is a sort of mainstream standard and has been drilled into our minds since babies.
To perform the diagonalization, we can simply look at each real number that you randomly generate and add digits to the diagonalized number based on that order. It doesn't matter how much you shuffle them around afterward: the diagonalized number will still be different from all the generated numbers.
1) I will produce an infinite amount of numbers, then stop
2) I will look at the infinite amount of numbers and then produce a number which is not within the infinite numbers
3) Hey look, the number I produced in 2) is not within the infinite set, so there must be a larger infinite set
The issue is that you can never finish 1). That’s the error/illusion.
But if you get to cheat, then I can also add a step 4) As long as you add digits to my numbers to generate an additional number, I will reorder the set of generated real numbers and assign each of them a natural number.
Of course the order will change from iteration to iteration. But in this way the generated real numbers will always have a corresponding natural number.
> Of course the order will change from iteration to iteration. But in this way the generated real numbers will always have a corresponding natural number.
The real numbers that you generate will always have a corresponding natural number. But the diagonalized number will never have a corresponding natural number, no matter how much you shuffle your generated values.
That means that you stop. Which implies you are not going to infinity.
If you are going to infinity, there is no side that “wins” (real or naturals). You can always keep generating an additional one for either side.
The same way, we could continue this discussion forever if we just want to keep going back and forth (although HN does impose a limit of replies). And you could only make an assertion about the length of each side of the conversations, after the conversation ends. But if the conversation keeps going forever, you could never say to talked more than the other. Unless you specify a finite range for which you want to measure.
Naturals and Reals are both infinite. Saying that the cardinality of one is different than the cardinality of the other, is just comparing the ideas of cardinality about them that we have come with, but their actual cardinality can never be known.
Where, in all my comments, do I say that we must stop counting? You're putting words in my mouth here.
Since we don't stop counting, we never know the diagonal number's full value (unless our order is predefined). All we know is that the diagonal number exists, and it is different from all the real numbers we will ever count.
> If you are going to infinity, there is no side that “wins” (real or naturals). You can always keep generating an additional one for either side.
Of course. I am not disputing this. If we count through two different sets in the same way, then the subsets of values we count through will have the same cardinality (equal to the cardinality of the set of natural numbers).
But there's an important distinction. If we count through the natural numbers one-by-one without stopping, it's possible to exhaust the whole set of natural numbers, such that every single number will eventually get counted to.
Meanwhile, if we count through the real numbers one-by-one, it is impossible to reach all of them. There will always be a bunch of real numbers that we will never ever count to. That's why we say that there are more real numbers than can be counted: there are uncountably many.
You didn’t say it explicitly. But you are implying it when you assume you can count the naturals.
By assigning a label (eg. aleph-null) to infinity, is assuming you can finish counting them. The only way to finish is by stopping.
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.
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.
That is why ascending order is the norm for the positive integers. For any positive integer X, it is clear that we will reach it in X steps using that generator function. So, it’s not a matter of being mainstream, it’s just the simplest.
The difference between the infinity of the reals and the positive integers is that there is no such generator function for the reals. No matter what function you give me, I can write a finite number that you will not reach in a finite number of steps.
Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps. On the other hand, try giving me a finite positive integer that I cannot reach in finite steps.
Reals and naturals both have the same number of symbols: unknowable (“infinity”).
> Feel free to give me a generator function and I will give you a finite number that it cannot reach in finite steps.
f(n)=n
If you give n to that formula, it will generate n in 1 step.
> On the other hand, try giving me a finite positive integer that I cannot reach in finite steps.
Can’t come up with one. As long as is finite you can always just output the number itself.
Anything finite is countable. Anything infinite is unknowable.
F(x)=x, will give you x in one step.