this post was submitted on 23 Oct 2024
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It's called countable and uncountable infinity. the idea here is that there are uncountably many numbers between 1 and 2, while there are only countably infinite natural numbers. it actually makes sense when you think about it. let's assume for a moment that the numbers between 1 and 2 are the same "size" of infinity as the natural numbers. If that were true, you'd be able to map every number between 1 and 2 to a natural number. but here's the thing, say you map some number "a" to 22 and another number "b" to 23. Now take the average of these two numbers, (a + b)/2 = c the number "c" is still between 1 and 2, but it hasnโt been mapped to any natural number. this means that there are more numbers between 1 and 2 than there are natural numbers proving that the infinity of real numbers is a different, larger kind of infinity than the infinity of the natural numbers
Your explanation is wrong. There is no reason to believe that "c" has no mapping.
Edit: for instance, it could map to 29, or -7.
Yeah, OP seems to be assuming a continuous mapping. It still works if you don't, but the standard way to prove it is the more abstract "diagonal argument".
But then a simple comeback would be, "well perhaps there is a non-continuous mapping." (There isn't one, of course.)
"It still works if you don't" -- how does red's argument work if you don't? Red is not using cantor's diagonal proof.
Yeah, that was actually an awkward wording, sorry. What I meant is that given a non-continuous map from the natural numbers to the reals (or any other two sets with infinite but non-matching cardinality), there's a way to prove it's not bijective - often the diagonal argument.
For anyone reading and curious, you take advantage of the fact you can choose an independent modification to the output value of the mapping for each input value. In this case, a common choice is the nth decimal digit of the real number corresponding to the input natural number n. By choosing the unused value for each digit - that is, making a new number that's different from all the used numbers in that one place, at least - you construct a value that must be unused in the set of possible outputs, which is a contradiction (bijective means it's a one-to-one pairing between the two ends).
Actually, you can go even stronger, and do this for surjective functions. All bijective maps are surjective functions, but surjective functions are allowed to map two or more inputs to the same output as long as every input and output is still used. At that point, you literally just define "A is a smaller set than B" as meaning that you can't surject A into B. It's a definition that works for all finite quantities, so why not?