Counting real numbers

[Anonymous No.9390680]

How do you start to count real numbers?
Do I start with 0.1, 0.01, 0.001, 0.0001 ....etc.

[Anonymous No.9390682]

The set of real numbers are uncountably infinite, unlike the set of all integers, which are countably infinite. There is a proof for it but I am on mobile rn. Just look up countably infinite sets on Google and you should get a clearer definition

[Anonymous No.9390738]

>>9390682
>phoneposting
kys Reddit

[Anonymous No.9390741]

>>9390738
>not phoneposting
gg granpa

[Anonymous No.9390742]

>>9390680
Simple, just find an algorithm for counting the Cantor Set, and then utilize a bijection from the Cantor set to the real numbers.

[Anonymous No.9390758]

>>9390680
You can start to count the Reals anyway you want, but any way you do it you won't be able to get all of them listed.

[Anonymous No.9390764]

>>9390738
i see people make this comment, and assume it's a joke i'm not getting. what's the story here..

[Anonymous No.9390791]

No, you start with

1, 2, 3, 4, 5, 6, 7, ...

When you're finished with these, come back to me and I'll explain how you can count the rest of the reals :^)

[Op No.9390982]

>>9390791
Ok explain

[Anonymous No.9390984]

>>9390680
uncountable, unlistable

https://youtu.be/elvOZm0d4H0?t=4m

[Anonymous No.9392282]

>>9390680
you use a slider u goof

[Anonymous No.9392285]

Put them on a well order and start counting.

[Anonymous No.9392971]

>>9390764
t.brainlet

[Anonymous No.9392985]

>>9390680
You make a bijection between the powerset of the natural numbers and the real numbers, and then you start counting them in order of increasing set size. That is {} {0} {1} {2} ... {0,0} {0,1} {0,2} ... {1,0}{1,1} {1,2} ... {2,0} ... {0,0,0} ... etc.

[Anonymous No.9393000]

>>9390680
The real numbers are not well-ordered under the standard ordering. The claim that they can be well-ordered is equivalent to the axiom of choice.

[Anonymous No.9393364]

>>9390680
To count the real numbers, you need to put them in a list.
Say you have a list of all the real numbers, in whatever order you want. We're going to write the numbers in decimal notation, and we'll remove all the numbers which are not between 0 and 1 for now. We also have to be careful not to write any numbers with infinitely many 9's trailing at the end, since we can just write these numbers with infinite zeros. In fact, just throw zeros on any terminating decimal. Our list is still valid.
For example, our list could start:
0.5823...
0.4821...
0.5000...
0.8917...
And on and on.

Let's think and see if we missed a number on our list.

For instance, what if there was a number whose first decimal place was different from the first decimal place of the first number in our list? In the list I made, our real number has something that isn't 5 in the first position.
0.2...

Now what if it has a different 2nd place from the 2nd place of the second number? We can't pick 8, so I'm going to now pick 0.26...

Continue this and get some number, in this case I'm constructing 0.2613... which is pretty random but you can construct such a number systematically (if 1, change to 2, if not 1, change to 1)

But this number can't be anywhere on the list! Because it's different from the first number by at least one decimal place, and from the second number, and the third, and so on, all the way down the entire list. It's not any of the numbers on the list, so we missed one number. In fact, we were supposed to have all the numbers between 0 and 1 to begin with - but we were missing a ton - so that's it. Our list is always going to be incomplete.
Since that's just an interval of the real numbers, you definitely can't count the whole set!
This method is called Cantor's Diagonal Argument.

[Anonymous No.9393420]

>>9390680
>real numbers

[Anonymous No.9394694]

[Anonymous No.9394766]

>>9390680
Prove the continuum hypothesis and then use cardinal numbers.