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/sci/ - Science & Math

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5971501 No.5971501 [Reply] [Original]

Are the reals not countable? What is wrong with my proof?

Proof:

Let n=0. Every time you think of a particular real number r, let n=n+1 and map r to n.

>> No.5971521

let's count the reals, shall we?

0.1 is real #1
0.2 is real #2
oh wait, there's another real between those two
ok, so
0.15 is real #2
oh wait, i forgot yet another one
0.11 is real #2
no, wait, forgot about this one
0.101 is real #2
...

>> No.5971535

>>5971521
bro, you don't need to count them in their "<" order

>> No.5971538

>>5971521
let's count the rationals, shall we?

0.1 is rational #1
0.2 is rational #2
oh wait, there's another rational between those two
ok, so
0.15 is rational #2
oh wait, i forgot yet another one
0.11 is rational #2
no, wait, forgot about this one
0.101 is rational #2
...

>> No.5971547

>>5971538

the rationals are countable
the reals are not
but I bet you already knew that

>> No.5971542

>>5971535
make a new real by diagonalising, it cannot be on your list

>> No.5971543

>>5971501
You didn't show that your map is well-defined, injective, and surjective. That's what's wrong with your proof.

>> No.5971554

>>5971535
if you can't count them while ordered you can't count them at all

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

end of thread, any further posts are bait

>> No.5971568

>>5971547
>>5971554
But by >>5971521's reasoning the rationals wouldn't be countable as demonstrated by >>5971538

>> No.5971571

>>5971543
>well-defined,
I defined it.

>injective,
With every real number the counter is increased, so there cannot be two real numbers with the same n.

>and surjective
The naturals are a subset of the reals, so the map has to be surjective.

QED

>> No.5971575

>>5971568
let's count the rationals shall we?

1/1
1/2
1/3
...
phew, that seemed to take forever! ok, now for the next batch!
2/1
2/2
2/3
...

>> No.5971586
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5971586

>>5971571
>injective,
>With every real number the counter is increased, so there cannot be two real numbers with the same n.

>> No.5971615

>>5971575
You cannot do anything AFTER infinity. 8/10 if troll

>> No.5971647

>>5971586
Where am I wrong? Whenever a new number is counted, the n takes the next value.

>> No.5971661

>>5971535
Every set can be well-ordered.

Put the reals in a well-ordering, then count them like that.

>> No.5971669

>>5971661
Every set is countable by the well ordering theorum?

>> No.5971743
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5971743

http://youtu.be/tXhtYsljEvY

I stopped caring about math when I was introduced to the concept of real numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't exist, like some kind of math deity , then you are fucking wrong and the math is flawed.

>> No.5971750
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5971750

http://youtu.be/ScLgc_98XxM

Same for constructive reals that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

>> No.5971755

>>5971542
I always felt like the diagonal argument could be abused to include numbers that "aren't on the list YET" and could then be applied to other sets that we know are countable.

>> No.5971773

>>5971750
If you ever studied algebra you'd know that notation like <span class="math">\sqrt{2}[/spoiler] is merely notation for a root of the polynomial of <span class="math">x^2 - 2[/spoiler] in some field extension of Q (which is proved to exist). An algebraist cannot distinguish between the two solutions, but in analysis you prove there is a distinct one with is > 0, and that is taken to be the default value when such notation is used.

Appropriate generalizations are left to the reader.

>> No.5971907

>>5971773
That dude in the guy's videos makes this distinction in some other videos.
Basically says there's three ways to talk about the square root of 2.

One is algebraic, one is analytic, and one is approximate. He's fine with algebraic but flips out about analytic and claims that algebraic isn't really a number and so only approximate exists as a number. He's actually kind of batshit because he goes on to make claims like "If you say that square root of 2 exists then you are creating a number and only god can create numbers. You are making yourself equal to god".

Anyway, arguing by talking about the algebraic field extension is kind of a moot point since he's cool with that. He only goes on to point out that in the algebraic method it's not really a number in the set.

Either way, all of his insanity aside, his constructions for linear algebra and trigonometry without using rationals do seem interesting and useful, even if unconventional.

>> No.5971909

>>5971907
I forgot to mention.

This guy is an obvious troll.
>>5971743
>>5971750
He posts this stuff in a lot of threads.

>> No.5971920

COPYPASTA EVERYWHERE
And people are fucking buying it.

>> No.5972331

this is not a correct definition since you have to determine which <span class="math">r[/spoiler] will be the next.
obviously two persons can get two different map using you definition aka it's not well-defined.
now get off /sci/ im watching quality threads die for this shit