Cantor revisited

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.

Other urls found in this thread:

lmgtfy.com/?q=cantor's original diagonal proof paper
twitter.com/SFWRedditImages

you're confused
read the proof again, carefully

touche'

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.

> cannot be ordered with a finite algorithm
Shows reals cannot finitely/algorithmically be ordered with THAT particular method and interpretation.

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.

>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

If you think it's crap, simply form a bijection between the integers and reals; you'll earn a Fields Medal

> 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.

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.

> just a picture for intuitive explanation
??? just ???
It's "just" a picture that Cantor himself wrote an 1891 paper about and generates @100,000 hits on google.

So what's your point ?

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.

>Cantor's proof merely shows the reals cannot be ordered with a finite algorithm.
What an idiot. The attempted list of reals is infinite.

You're either replying to trolls or replying to retards that think their opinions have mathematical validity. Don't waste your time with carefully thought responses, mate.

Be so kind as to provide a source for the
various original proofs that are not buggered up by some institution translation or in the original German
language.

Can you fucking not use Google or something?

lmgtfy.com/?q=cantor's original diagonal proof paper

>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.
No, you can't.

Why are YOU here troll ?
Don't think people can make up there own mind on whether to talk about this area of mathematics ?

Oh yes you can, but it won't mean anything more than power set proofs.

Let A be a set. Let P(A) denote the set of subsets of A. We will show there's no bijection from A to P(A).

Assume on the contrary that there exists a bijection f:A -> P(A). Now take

[math]B := \{ a \in A / a \notin f(a) \} [/math]

Let a be such that f(a) = B. If a is in B, then by the definition of B a is not in f(a) = B, a contradiction. If a is not in B, then by the definition of B a is in f(a) = B. a contradiction. Therefore B is not in the image of f, and f can't exist.

...

Provide a direct link if it's so easy faggot.

no, you can't. if you think you can then fucking go ahead and construct it, you ignorant, pedantic piece of shit

If you don't have the ability to go through a couple of results on Google to find a fucking paper, then it's highly unlikely you have the ability to even get past the first few sentences of the proof, limp-dicked piece of shit.

>now if you look closely, you can see that 1 is not in that sequence
you're misunderstanding the proof. you have to form a bijection between the two sets. so take the natural numbers and the natural even numbers and form a bijection such that 1 maps to 2, 2 -> 4, 3 -> 6, etc. no matter what natural number, N, you think of, it will map to a number in the set of positive even numbers which will be equal to 2N. You can't find a number in the set of all natural numbers that does not map to a number in the set of all positive even numbers therefore they are equal. When you form a bijection between the reals and the naturals though, you can construct a real number that does not map to a natural number. not only that, but you can construct an infinite number of such numbers. therefore the reals are bigger than the naturals.

...

>Therefore B is not in the image of f, and f can't exist
because P(A) can't exist? wildberger was right all along, the reals aren't real. math is a lie.

my proof is under standard ZFC, where Power Set is an axiom. I don't think wildberger denies PowerSet, but Infinity. Infinity says N is a set.

>so how can you be so sure intuitive proof by contradiction rules still apply there ?
>There is no finite complete list of natural numbers
it's a proof about infinite sets. if you assume the existence of infinite sets, it's valid. If you don't, then why would you be wondering about the properties of infinite sets in the first place?
>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
try it

> it's a proof about infinite sets. if you assume the existence of infinite sets, it's valid.
It may be valid in terms of a (arbitrary ?) set of axioms using mathematical principles not originally conceived to handle infinity.

> If you don't, then why would you be wondering about the properties of infinite sets in the first place?
Reject infinite sets altogether? It's fairly easy to define and discuss an infinite set - 2 3 5 7 11 13 ... for example.

the problem is you never studied serious mathematics
it's not "fairly easy to define and discuss an infinite set". "..." doesn't mean anything. formalizing infinite sets to work like we want them to needs a special set of axioms

Ad hominen attack. mmmm.

Fine this thread is henceforth for those who have not spent there life twiddling with their peano postulate.

I'm sorry, you'll have to find a thread with more Latex in it.

Thanks for dropping by though.

Goodbye.

which peano postulates? the peano axioms for N are not a set theoretic construction. rigorous construction of N from sets needs much more.

you're also a fucking autist. why the fuck does someone come on Veeky Forums and think making a post like this is a good idea? you look like a complete autist user, fucking cut it and talk like a real person.

gibe da medl b0ss

I would tell you that you should educate yourself a little more, watch less Youtube videos and get off the internet from time to time, maybe open up a textbook, but since you can't separate your feelings from coherent discussion and seeing that you lack the basic cognitive skills to differentiate 'there' from 'their', those are definitely not options for you. May I suggest suicide or social work as an alternative?

Here's the issue:

Can you reach any real number in a finite number of steps?

I think it's pretty obvious that's not going to be the case based upon the decimal expansion of the real numbers.

the fact that you have the articulative power of an autistic vietnamese peasant doesn't raise any suspicion or self-doubt that maybe you don't have the grip you think you have on the material you are discussing?

> Can you reach any real number in a finite number of steps?

If you answer no a paradox is created.

That two entities can never independently choose the same transcendental number.

For example, if I choose PI you are "not allowed" to ever choose the same number, because if you did that would be reaching agreement in a finite number of steps.

What the fuck are you talking about?

This thread is too juicy!

>wut?

>these sick burns are almost too much for/sci/

>not taking your pills before going on the internet

>never come back plz

>"not going to be the case based upon the decimal expansion"
give yourself a cookie for that one

>insulting other idiots for them being idiots

>It may be valid in terms of a (arbitrary ?) set of axioms
all axioms are arbitrary.
>using mathematical principles not originally conceived to handle infinity.
what a load of crap. most of mathematics was not originally a purpose for it. that doesn't mean it's invalid. infinite sets are provable in various set theories. If they create no contradiction, they are valid, regardless of what was originally conceived when the axioms were set.
>Reject infinite sets altogether?
you missed the entire point of that sentence, read it again
>it's fairly easy to define and discuss an infinite set - 2 3 5 7 11 13 ... for example.
Only some poor philosophical discussion, won't be in rigorous discussions of math if you do away with cantor's argument. you can't easily keep the idea of infinite sets while invalidating cantor's argument in math. certainly not in ZFC.

If I put 1 after all the even numbers, it still works. The first set has cardinality aleph0 and so does the second one.
Cantorfags btfo

>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 the procedure that generates digits of e=2.7182818... doesn't really halt.
Most people call that a well defined algorithm.

When you numerically calculate e, you specify the number of digits you want, and then stop when you hit your target. If you can show your method converges, you can show this happens after a finite number of steps. Thus it's an algorithm.

>Most people call that a well defined algorithm.
I don't.

I'd love to see you run a never ending algorithm.

In fact run a while true loop on your computer and I assure you it produces all digits of pi after the loop is complete.

> I'm not even sure what you mean by an infinite algorithm
One complaint you here on Cantor's diagonal proof is that the diagonal line is allowed to never halt
but the list is kind of all there, finite in concept at least, and ready to be ravaged by Cantor's
merciless diagonal.

The algorithm itself is non-halting, in practice you either run out of time or printer paper.

These posts are lame, even for fools like you.

so in practice Cantor's diagonal doesn't exist because you run out of time or printer paper.

Thanks for admitting that.

What about 1/3? This does not even cover all rationals.

there is no requirement that an algorithm halt. If there was, the halting problem wouldn't be a problem in the first place. it'd be true by definition.

that's like saying the reals aren't infinite because you can only write a number so big.

exactly, thank you for making my point.

I require a valid algorithm to halt. Every reasonable person does.
The halting problem only occurs because we allowed mathematics to go down a really bad road.

In the context of your sick twisted homosexual analogy it would be more like Cantor demands the list runs out of paper before his diagonal runs out of paper.

and I demande my
while true
print 1

to run out of paper before it can produce all the digits of pi.

oh sorry i forgot math and computer science are invalid the moment they become unintuitive.to the average retard. Fuck axioms, just base it all on subjective definitions of common sense.

If this actually means anything rephrase.

>oh sorry i forgot math and computer science are invalid the moment they become unintuitive.to the average retard
they're not unintuitive, they're just inferior mathematics (less constraints, and ironically less realistic)

for some reason, the other guy demands an infinit list of numbers before applying the argument of Cantor. That list would make us "run out of paper".
I just do the same by asking that my (ridiculous) algorithm prints "1" indefinitely, making us run out of paper, and I then claim that it would produce all the digits of pi after that. It's just something to show his argument is ludicrous.

So where is the OP violating any axioms ?

does a harmonic analogon have something to do with this?

complete math noob, im here to learn

>they're just inferior mathematics
unorganized mathematics is superior to axiomatic mathematics?
where did I say he was? are you even reading what you're replying to or are you just jumping into discussions cause you're a sperg and think someone said something you don't like.

>unorganized mathematics is superior to axiomatic mathematics?
no, mathematics with pertinent axioms is superior to mathematics with pointless axioms.
Jesus christ if you're this retarded, no wonder you just choke on cantor's cock instead of thinking for yourself.