/sci/ - Science & Math » Thread #13955019

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Anonymous Sun 12 Dec 19:07:00 2021 No.13955019 View Reply Original Report

Quoted By: >>13955025 >>13955047 >>13955137 >>13955153 >>13955188 >>13955213 >>13956266 >>13956637

Can someone prove this for a brainlet?

Anonymous

Anonymous Sun 12 Dec 2021 19:10:03 No.13955025 Report

Quoted By:

>>13955019
5 minutes ago this board was infinitely shit, now it's infinitely shittier.

Anonymous

Anonymous Sun 12 Dec 2021 19:12:18 No.13955031 Report

Quoted By:

Look up Cantor's theorem, that

For any set X, there is no injection from the power set of X into X.

Similarly one can consider Rotnac's theorem:

There is no surjection from a set to its power set.

Robin Knight provides a good proof of both of these in his 2020 set theory notes.

Anonymous

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Anonymous Sun 12 Dec 2021 19:15:19 No.13955038 Report

Quoted By:

>bigger

Anonymous

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Anonymous Sun 12 Dec 2021 19:20:20 No.13955047 Report

Quoted By: >>13955392 >>13955400 >>13955403 >>13956372 >>13956606 >>13956617

>>13955019

Natural numbers: 1, 2, 3, ...
There are an infinite number of them

rational numbers: X/Y
ex 1/2, 2/3, 2/7, 8/9, (4/2 = 2) etc.
There are an infinite number of them AND they include all the natural numbers so they are a BIGGER infinity than natural numbers

Now for brain explosion.
You can count (1 to 1 matching) all the rational numbers using the natural numbers

When dealing with infinities your concepts of bigger and smaller need to be 'expanded'

Anonymous

Anonymous Sun 12 Dec 2021 19:51:48 No.13955137 Report

Quoted By:

>>13955019
Yes, limit x-> inf x/x^2 = 0
Or
There are more real numbers between 0 and 1 than there are integers

Anonymous

Anonymous Sun 12 Dec 2021 19:54:59 No.13955153 Report

Quoted By: >>13955173 >>13955192

>>13955019
A set of infinite sets will always be bigger than a set of infinite variables.

Anonymous

Anonymous Sun 12 Dec 2021 20:00:46 No.13955173 Report

Quoted By: >>13955185

>>13955153
I do not know what an 'infinite variable' is. But if there is at least one, then the set O of infinite
sets is surely smaller than any nonempty set of 'infinite variables'.

Anonymous

Anonymous Sun 12 Dec 2021 20:04:10 No.13955185 Report

Quoted By:

>>13955173
Note to self: do not use UTF characters on 4chan.
I meant to type $\emptyset$

Anonymous

Anonymous Sun 12 Dec 2021 20:04:42 No.13955188 Report

Quoted By: >>13955274

>>13955019
Cantor's theorem states that there is an injection X to P(X) but no bijection between the two.
I don't think there's a nice proof that there isn't an injection P(X) -> X that doesn't use Cantor-Schroder-Bernstein.

Anonymous

Anonymous Sun 12 Dec 2021 20:06:01 No.13955192 Report

Quoted By:

>>13955153
{N} is a set of infinite sets, and it's not bigger than the set {x_infinite, x_anotherinfinite} of infinite variables.

Anonymous

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Anonymous Sun 12 Dec 2021 20:11:17 No.13955213 Report

Quoted By: >>13955225

>>13955019
infinity + 1

Anonymous

Anonymous Sun 12 Dec 2021 20:13:49 No.13955225 Report

Quoted By:

>>13955213
Kek

Anonymous

Anonymous Sun 12 Dec 2021 20:27:50 No.13955274 Report

Quoted By:

>>13955188
>Cantor's theorem states that there is an injection X to P(X) but no bijection between the two.
Indeed.
>I don't think there's a nice proof that there isn't an injection P(X) -> X that doesn't use Cantor-Schroder-Bernstein.
You are confused. It is the statement that P(N) has the same cardinality as R which requires
Cantor-Schröder-Bernstein. It has nothing to do with Cantor's original statement.
Anyway, Dénes K?nig supplied an easy proof of Cantor-Schröder-Bernstein using Graph Theory.
It is not like it would be something that a first year CS student couldn't do.

Anonymous

Anonymous Sun 12 Dec 2021 20:58:57 No.13955392 Report

Quoted By: >>13955400

>>13955047
Even numbers: 2, 4, 6...
There are an infinite number of them

Natural numbers:
ex 1, 2, 3, 4 etc.
There are an infinite number of them AND they include all the even numbers so they are a BIGGER infinity than even numbers

Anonymous

Anonymous Sun 12 Dec 2021 21:00:24 No.13955400 Report

Quoted By: >>13955454 >>13955519 >>13956343

>>13955392
>>13955047

Are all countable sets "equal" in size?

Anonymous

Anonymous Sun 12 Dec 2021 21:01:01 No.13955403 Report

Quoted By:

>>13955047
>You can count (1 to 1 matching) all the rational numbers using the natural numbers
>they are a BIGGER infinity than natural numbers
Is this supposed to be trolling?

Anonymous

Anonymous Sun 12 Dec 2021 21:15:27 No.13955454 Report

Quoted By:

>>13955400
>Are all countable sets "equal" in size?
If there's a one-to-one mapping between the elements, how can there be any less or more elements in one set than in the other?

Anonymous

Anonymous Sun 12 Dec 2021 21:32:32 No.13955519 Report

Quoted By: >>13956338

>>13955400
do you mean to ask if all countably infinite sets are equal in size? because you can demonstrate that countable sets are not necessarily equal in size if you compare for example the set of all primes from 0-100 and the set of all odd numbers from 0-100

Anonymous

Anonymous Mon 13 Dec 2021 00:43:46 No.13956266 Report

Quoted By: >>13956665

>>13955019
1*0=0, 2*0=0. Logically, $\frac{1*0}{2*0} = \frac{0_1}{0_2} = \frac{1}{2}$ should be true, meaning some zeroes are "bigger" than others

infinity is just 1/0, so the same logic should hold.

Anonymous

Anonymous Mon 13 Dec 2021 01:08:38 No.13956338 Report

Quoted By:

>>13955519
>do you mean to ask if all countably infinite sets are equal in size?

Yes, I thought we were only talking about infinite sets, but I should have been more precise.

Anonymous

Anonymous Mon 13 Dec 2021 01:11:11 No.13956343 Report

Quoted By:

>>13955400

Is there a "smallest" infinite countable set and a "largest" infinite countable set?

Anonymous

Anonymous Mon 13 Dec 2021 01:21:23 No.13956372 Report

Quoted By:

>>13955047
>Natural numbers

There are no numbers in nature

>rational numbers: X/Y
so there can't be "rational" numbers too.

>>13955047
>When dealing with infinities your concepts of bigger and smaller need to be 'expanded'
You can think about endless versions of infinity because they did not really exist. Delusions generated by an overdose of mathematics.

Anonymous

Anonymous Mon 13 Dec 2021 01:31:42 No.13956409 Report

Quoted By:

no, you need to tell me the axioms I have available first
then maybe I'll tell you the proof
give me the axioms
I NEED THE AXIOMS

Anonymous

Anonymous Mon 13 Dec 2021 02:40:55 No.13956606 Report

Quoted By:

>>13955047
>count 1 to 1
okay, let me know when youve done all the 1 to 1 counting.

Anonymous

Anonymous Mon 13 Dec 2021 02:44:00 No.13956617 Report

Quoted By: >>13956642 >>13956644

>>13955047
infinity isn’t a number, it’s a concept and as such you can’t count any of it
there are no bigger or smaller infinities
and does a set of all sets include itself

Anonymous

Anonymous Mon 13 Dec 2021 02:49:37 No.13956637 Report

Quoted By: >>13956654

>>13955019
Watch this, it's a crayon&hand puppets level intro to infinities.
https://youtu.be/SrU9YDoXE88?t=1m

Anonymous

Anonymous Mon 13 Dec 2021 02:50:39 No.13956642 Report

Quoted By:

>>13956617
lolno

Anonymous

Anonymous Mon 13 Dec 2021 02:51:13 No.13956644 Report

Quoted By:

>>13956617
numbers are concepts you low iq banana

Anonymous

Anonymous Mon 13 Dec 2021 02:54:16 No.13956654 Report

Quoted By:

>>13956637
https://www.youtube.com/watch?v=i7c2qz7sO0I
https://youtu.be/elvOZm0d4H0?t=2m
https://www.youtube.com/watch?v=BBp0bEczCNg
https://www.youtube.com/watch?v=FVZqPaH94qU

Anonymous

Anonymous Mon 13 Dec 2021 03:00:08 No.13956665 Report

Quoted By:

>>13956266
what is lim_x--> 0-

Anonymous

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Anonymous Mon 13 Dec 2021 03:20:54 No.13956734 Report

Quoted By:

And there you go

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