Cantor's proof merely shows the reals cannot be ordered with a finite algorithm.
There is no proof of "different types of infinity".
There is only the one familiar infinity { 12345678...INFINITY } that everyone already knows about.
Cantor revisited
Ethan Jenkins
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Jaxson Hughes
you're confused
read the proof again, carefully
Ryder Butler
touche'
Jack Lopez
cantor's proof is like me saying:
take the sequence of even natural numbers 0,2, 4 etc
Now if you look closely, you can see that 1 is not in that sequence. So there must exist a set bigger than N.
when in reality you just "constructed" a new natural number.
Josiah Taylor
> cannot be ordered with a finite algorithm
Shows reals cannot finitely/algorithmically be ordered with THAT particular method and interpretation.
Connor Walker
All of you seem to have major gaps in your understanding of what Cantor said.
>Cantor's proof merely shows the reals cannot be ordered with a finite algorithm.
No, that's not what Cantor showed. Cantor demonstrated no algorithm can exist. The argument is a proof by contradiction. He supposes that you posses an algorithm that can fully enumerate the reals as an ordered list, and then shows that no such algorithm could exist by constructing a real outside the ordered list. He makes no assumptions about the nature of the algorithm.
Besides, I'm not even sure what you mean by an infinite algorithm. You mean one that never halts? Because if you ask most people they won't call that an algorithm. It's gotta stop and produce a result eventually to be an algorithm.
>So there must exist a set bigger than N.
That's not how infinite sets work. For instance, the set of all integers and the set of all even integers are the same size, namely [math]\aleph_0[/math]. This may seem to be counter intuitive because it seems like there are twice as many integers as even numbers, but little is intuitive once you start considering infinities. The key observation here is that they are both infinite sets, and that you can form a bijection between them and the integers (read "count them").
No algorithm is presented in the proof, it merely supposes one exists. Nothing is assumed about the nature of this algorithm, just that it can produce an ordered list of the reals.
The difference between countably infinite sets (like the integers) and uncountably infinite sets (like the reals) is a very important distinction. If you think it's crap, simply form a bijection between the integers and reals; you'll earn a Fields Medal. While you're at it, I think you will find Cantor's argument more convincing.
Carter Bennett
>If you think it's crap, simply form a bijection between the integers and reals; you'll earn a Fields Medal
Rofl, are you actually under 40? And you're trying to lecture me?
Silly undergrad pleb wasting his precious days on Veeky Forums
Lucas Lee
If you think it's crap, simply form a bijection between the integers and reals; you'll earn a Fields Medal
Angel Jones
> shows that no such algorithm could exist by constructing a real outside the ordered list
But, that proposed list of ordered reals is an infinite list, so how can you be so sure intuitive proof by contradiction rules still apply there ?
There is no finite complete list of natural numbers. So under the reasoning of the Cantor proof, one could theoretically construct
a similar proof that shows a special infinite set of naturals cannot be put in one to one correspondence with the complementary set of naturals.
Christopher Roberts
you don't need a "list", that's just a picture for intuitive explanation
read the algebraic proof and you'll understand it. a picture is not a proof. read the proof.
stop posting otherwise.