Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers.
I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem.
And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.
I think "countably infinite" makes no sense to you because you have a different idea of countable than a mathematician (disclaimer: not a mathematician).
A mathematician compares the size of two sets of stuff by pairing off items from each set, but this is not a mechanical process taking a finite or even unbounded amount of time: they just need to show such a mapping exists or that nonexistence would lead to a contradiction; they don't need to actually carry out the process mechanically. By definition (according to mathematicians), something is countable if it is the same size as the set of natural numbers {0, 1, 2, ...} or smaller, and "countably infinite" just means it is the same size as the naturals (and not smaller, which would make it finite).
A small minority of mathematicians hold the position that proof-by-contradiction is not good enough, and that you really do need to positively prove something. They are called intuitionists.
Presumably, an even smaller minority of mathematicians hold the position that this proof must (theoretically) be able to be carried out in a mechanical manner. They're some flavor of constructivists, but maybe they're better called programmers. <- This is where you are.
> Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers.
Exactly correct! This holds true of everything you can generate stepwise, even infinite sets. Cantor proved that you cannot "generate" (stepwise) all Reals between 0 and 1. Any infinite set you can generate stepwise is Countably Infinite.
> I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem.
Exactly correct! And then infinitely later, when you're "done", having generated every Real between 0 and 1, he will then generate a new Real not on your list. Oops! You have not generated all Reals between 0 and 1, even with infinite time.
> And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.
Exactly correct! Any two sets you can pair together (via a bijection) have the exact same cardinality. Neither is more infinite nor countable than the other. Cantor proved you cannot "pair" the Reals with the Natural Numbers.
You and Cantor agree completely. You're very close to understanding why the Reals are bigger.
There can be no 'and then' after infinitely later.
I don't see why stepwise is important but that must be the key to Cantor's proof.
If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned.
The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to match.
Is it that my black box metaphor is cheating by coercing a truly 'parallel' generation of decimals into a linear operation? But even then, if I'm getting exponentially bigger chunks of new decimals, I can provide equally large chunks of integers... so it still doesn't make sense to me. Infinity is infinity and you cannot count it.
Mathematicians consider two sets to be of the same size or more precisely "cardinality", if it is possible to construct a 1-1 map of elements from the first set to the second set. These maps can obviously be constructed for sets with finitely many elements, and they can be constructed for sets with an infinite number of elements as well. For instance, the set of all integers has the same cardinality of the set of all positive integers (just enumerate the integers alternating back and forth expanding from 0 - this constructs the 1-1 map).
We can prove that no such 1-1 map exists between the integers (an infinite set) and the decimals in the interval [0,1] (another infinite set). The proof is by contradiction, meaning that we assume such a 1-1 map exists and prove it leads to a contradiction, therefore our assumption that the 1-1 map exists must be false.
So suppose we were able to construct a map from all decimals in [0,1], by enumerating them according to some clever rule. Let d_i be the ith digit of number i in your mapping. For each I pick another different digit d_i'. Let's construct the number with decimal representation D = . d_0' d_1' d_2' ...
Assuming we have our 1-1 map, it must be somewhere in our mapping. Let's say it's element k. By our labeling concention the kth decimal digit of D is actually d_k. However, this contradicts our method of construction of D. Therefore our assumption that there is a 1-1 map between decimals in [0,1] and the integers must be false.
It is in this sense that there are infinities of different sizes.
> It is in this sense that there are infinities of different sizes.
They aren’t actually different sizes, though.
All this proves is that under specific set theoretic assumptions, a contradiction arises if you define “size” as “cardinality” and assume that a particular bijective relation exists between your two infinite sets.
It doesn’t actually mean the sets have different sizes, it just means they differ under a set of assumptions that may (or may not) be useful for your purposes.
I don’t have one that doesn’t admit a contradiction here … which I’d argue is because comparing the size of an infinite set is nonsensical, even if the properties used to do so are otherwise useful.
Similarly, I can also work around Russel’s paradox by introducing infinite universes, but that doesn’t actually resolve the paradox, it just provides a set (ha ha) of rules that may be leveraged to formalize the Set category and otherwise prove useful things.
Just because your formalization admits a proof by contradiction doesn’t actually prove two infinite sets have different sizes, it just proves that a contradiction exists under your assumptions.
If you aren't allowed to operate in a logical system with a concrete definition of "size", then you can't say things like "doesn't actually prove two infinite sets have different sizes". So the whole debate is moot.
> If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned.
Exactly correct! Any bijection between the naturals and the reals would suffice to show that they're the same cardinality; the order does not matter. I think where you're getting confused is just in who's trying to do what; who's the "protagonist" and "antagonist" in the proof.
Cantor is not trying to overwhelm you with so many real numbers that you run out of integers. Instead, he completely accepts and agrees with everything you're saying. And then he says: okay, pick any numbering of the reals you like. 1.11 is 4, 1.111 is 76, and 1.1111 is 445662323. It doesn't matter. You pick the pairing. Write your pairing down on an infinitely long sheet of paper. If the reals and integers have the same cardinality, there must be some way to write them all down on an (infinitely long) list. Pick any one and write it down.
Cantor's only job now is to show you that any real number exists that is not on your list. To do this, he constructs a number a digit at a time. He looks at the 1st digit of the 1st number, and writes down a different digit for his 1st digit. He looks at the 2nd digit of the 2nd number, and writes a different one for his 2nd digit. He looks at the nth digit of the nth number and writes a different one down for that digit, for every digit. Real numbers never run out of digits, so this goes on forever.
If this number he has written down is on your list, you should be able to point to a number on your list and say "Aha! You see, that is just real # 65,334,649!" but you can't, because it's different from that number in its 65,334,649th digit. It is truly different from every number on your list. And so there are more reals than integers.
I feel the number he generates via any procedure would be on the list since all of them are on the list. What am I missing about the plausibility of a procedire that must be possible which must generate a real number not in the series?
> I feel the number he generates via any procedure would be on the list since all of them are on the list
The claim is that all of them are on the list. The constructed number proves that claim false.
It's a proof by contradiction. If you assume there is any way to write an infinite numbered list of all reals, then Cantor shows it's possible to come up with a number not on your list. The construction uses your list as input, and given any list, can always produce a real number not on that list. Therefore there is no way to write an infinite numbered list of all reals.
It relies on the fact that real numbers have (countably) infinite digits, and therefore infinite "degrees of freedom" to be different. This may be one reason it's hard to accept. A "true" real number can contain infinite information in a single number. For instance, we can jam all of the naturals into a single real by just concatenating their decimal representations: 0.1234567891011121314151617181920212223...
This one single real number encodes the full infinite natural number line. That hopefully gives you a sense of why the "infinite digits" definitions of reals makes them qualitatively "bigger" than any number that has finite representation.
No I still don't get it, it's like saying that infinity^2 is larger than infinity. If 0^2 is no larger than 0, then it must be the same for infinity.
I see how a list of reals is like 2D list of infinities, so one grows from the middle and the other grows from the end, but they're both still infinite. I guess I'm still stuck in a 'mechanical' approach and not a mathematical one. I'm not sure I want to leave ;) This has been fascinating to think about anyway.
Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers.
I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem.
And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.