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To clarify. There is no mapping from natural numbers to all the irrational numbers. And that’s why there are different sized infinities

• nico 4y ago
  Why not? There’s no case in which you can have a complete map anyways. If you say one infinity is larger/smaller than another, you’d be saying that the complete map for one is larger/smaller than for the other one. But if they are both infinite, you will never stop counting, so you just can’t know. I know the diagonalization argument. But reality is that you can’t even count natural numbers. So if you have finite time to count any set, the set will be finite, and so it’s countable. But if you have infinite amount of time, any set for which there’s a formula to produce an element of, will always produce infinite elements at whatever speed you are able to produce them. If one formula is faster than another (eg n=(n-1)+1 vs n=(n-1)!), then you will count more of one than the other in the same amount of time, but that is processing power, not the size of the sets.
  • haneul 4y ago
    You can count natural numbers. Elementary school children do it every day. Now, try to count the real numbers. What comes after 1?
    • nico 4y ago
      > What comes after 1? Whatever I want. Why do you have to produce them in some predefined order? You could generate random real numbers, one at a time, and sort them as you add them to your set. When you stop, you will have a set of real numbers in order. If you want to know where a new one goes, you just generate it, add it to the set and sort it (or just add it in the proper position). And you will have a finite set of real numbers. If you never stop, you will keep forever generating real numbers, exactly the same as you would with natural numbers. The only thing that matters is whether you stop counting or not. If you stop counting, you get a finite set. If you don’t stop counting, you get an infinite one (but you’d never finish generating it). There is no way for a human to count for infinite time… so we can only speculate about that case. The only case that will ever occur in reality for us, is the finite case.
      • LegionMammal978 4y ago
        > There is no way for a human to count for infinite time… so we can only speculate about that case. We can talk about whether or not a countably infinite sequence S contains a given element x. We iterate through each element s of S one-by-one. If x is contained in S, then eventually we will find an element s such that s = x. But if x is not contained in S, then we will keep iterating for all eternity. For all countably infinite sets, we can always produce such a sequence, where the iteration will eventually halt if and only if the element is contained in the set. But for the set of real numbers, no matter what sequence we choose, we will never ever find the diagonalized number in our sequence. That's why we call the set of real numbers uncountable. You dispute that we can't physically count through all of the infinite elements in the real world. But math has no problem talking about what would hypothetically happen if we were to try. It lets us prove ahead of time that certain events will eventually happen if we iterate long enough, and other events will never ever happen.
        • nico 4y ago
          > But for the set of real numbers, no matter what sequence we choose, we will never ever find the diagonalized number in our sequence. So then it’s only proving that if you choose the ordering ahead of time, you won’t be able to do it, because real numbers don’t have a predefined order. You can only order them as you produce them. If you re-sort after each iteration (or insert them in order), you can count as many real numbers as you want. In any case, for practical applications, you could never count anything infinite, at some point you’d run out of physical storage to keep the count. How many bits can be encoded in the universe? That will give the limit of what could ever be counted. And it’s not infinite.
          • hgsgm 4y ago
            We're talking about infinity, not practical considerations. > (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.
  • nmilo 4y ago
    Countable (roughly) means that you can count from one element to any other in a finite amount of time, it doesn't mean you can count the whole set in a finite amount of time.
    • nico 4y ago
      Think about counting as three operations: 1. Produce a number 2. Add it to the set of numbers I’m counting 3. Sort the numbers Usually when you produce a number, you produce it in order (eg. 1, 2, 3, 4). But you could also count like: 4, 1, 3, 2 and then sort the set. As long as you have an algorithm/function to keep producing numbers for “each iteration”, you can always keep counting one at a time. The case you say: “count from one element to any other” is equivalent to a finite operation, it has a very clear beginning and end. But an infinite set doesn’t, unless you stop counting.
      • cscurmudgeon 4y ago
        > Think about counting as three operations: > 1. Produce a number 2. Add it to the set of numbers I’m counting 3. Sort the numbers Missing in your definition is the operation has to produce all numbers that you are counting over. If you are counting something, you don't leave out elements.
        • nico 4y ago
          If you are counting a set that is larger than your count, you will always leave elements out. The only way you won’t leave elements out for an infinite set is if you keep counting forever.
          • cscurmudgeon 4y ago
            "If you are counting a set that is larger than your count" What does that mean?
            • nico 4y ago