/sci/ - Science & Math

[Anonymous No.14668151]

Pick a random real number, assign it to 1. Pick another random real number, assign it to 2. Repeat the process to infinity until all real numbers have been mapped. Since the natural numbers are infinite, there is no point at which a real number cannot be mapped to a natural number.

Therefore there exists a bijection between the reals and natural numbers, and between all infinite sets. So all infinite sets have the same cardinality.

Q.E.D.

[Anonymous No.14668196]

>>14668151
>Q.E.D.
wtf does this have to do with anything?

[Anonymous No.14668207]

If the diagonal argument works for the reals, then it also works for the rationals, since they can also be represented by infinite decimals. But the rationals are known to have the same cardinality as the naturals, therefore the diagonal argument is flawed.

[Anonymous No.14668245]

>>14668151
nigger

[Anonymous No.14669184]

>>14668151
Bssed

[Anonymous No.14669199]

>>14668151
Outstanding, now let's take your list and we can construct a real number different from every indexed real by adding 1 to the appropriate decimal place (or subtracting 1 in the case of 9), we have now found a real number a real number that is inequal to every indexed real in at least one place and hence inequal to every indexed real, hence your index is incomplete as it has to failed to account for the new real.

[Anonymous No.14669214]

>>14669199
>we can construct a real number different from every indexed real
You can't

>>14668151
>Repeat the process to infinity until all real numbers have been mapped

As you see every number has been mapped already

>by adding 1 to the appropriate decimal place (or subtracting 1 in the case of 9)
You merely creating a number that will always show up in the list already

[Anonymous No.14669228]

>>14669214
>show up in the list already
by (valid) construction, our new number differs from any indexed number
Assume you've written down the list, then taking the diagonal, even if its infinitely large, produces a number that MUST be unequal to every member, even a 5 year old can see this

[Anonymous No.14669233]

>>14668151
this doesn't cover even the real numbers between 0.1 and 0.2

[Anonymous No.14669249]

>>14669228
>by (valid) construction, our new number differs from any indexed number
It doesn't, the premise is that you already listed all numbers, so no number can differ from the indexed. That could only happen in a finite list, but supposedly it's not a finite list.

[Anonymous No.14669283]

>>14669249
And the premise is absurd because of the diagonal argument
>That could only happen in a finite list
Not at all, I don't think you understand the construction.
Here's an infinitely long list that presumably contains all reals, by virtue of differing from each number in the list by at least one decimal place, you've constructed a number that, by definition, can't belong to the list even if you go on infinitely, as by construction, the number changes each iteration you go to account for the new members.
Keep in mind that the constructed real will obviously be irrational.

[Anonymous No.14669294]

>>14669249
Well your premise leads to a contradiction hence it should be discarded

[Anonymous No.14669318]

You can map rationals to a 2d space, because any rational can be written a / b, a and b are the coordinates.

Starting from natural numbers, how many coordinates do you need to specify pi ? Is there a function to generate those coordinates ?

[Anonymous No.14669322]

>>14669214
>You can't
why can't I?

[Anonymous No.14669341]

>>14668151
>Pick a random real number, assign it to 1. Pick another random real number, assign it to 2. Repeat the process to infinity
There is no way to do that.

[Anonymous No.14669365]

>>14669294
>Well your premise leads to a contradiction hence it should be discarded
It doesn't lead to a contradiction, you are the one contradicting yourself by claiming the list is infinite but treating it as finite

[Anonymous No.14669369]

>>14669322
>why can't I?
Because the number is already indexed at an arbitrary n position

Every time you add a digit to your "new" number imagine the number you currently just glows at some position of the list. This continuous endlessly as the premise is the list is infinite.

[Anonymous No.14669372]

>all the brainlets ITT offering invalid refutations to OP's invalid proposition
Cringe. >>14669341 gets it right.

[Anonymous No.14669379]

>>14669372
the only way to refute the real numbers is to be a finitist. the moment you concede that you can performe infinite actions then real numbers are automatically defined

[Anonymous No.14669387]

>>14669369
The premise is that the list is complete, you can't add any new numbers.

[Anonymous No.14669390]

>>14669379
Don't care. Crawl back to your hole, brainlet.

[Anonymous No.14669394]

>>14669369
You never add any digits. The premise of the list is that it's already complete.

[Anonymous No.14669395]

>>14669390
>no argument

[Anonymous No.14669398]

>>14669387
>The premise is that the list is complete, you can't add any new numbers.
Yes, then why infinitist retards even attempt to add a new number?

[Anonymous No.14669401]

>>14669394
The list is already complete, I'm just showing you the diagonalization argument is retarded because the supposedly new number that wasn't in the list IS in the list already

[Anonymous No.14669405]

>>14669395
You sharted out a completely incongruent reply. What am I supposed to be arguing against? The fact of the matter is that there's no way to "pick a random real number".

[Anonymous No.14669410]

>>14669398
>I have to prove real numbers have the same cardinality as natural numbers
>let's assume they have
>Q.E.D.

[Anonymous No.14669416]

>>14669398
They attempt successfully to construct a new number that can't belong to this "complete" list hence contradicting the fact that it's complete.
I'm quite confident my 9 year old niece could understand this, not sure why it's so difficult for you.

[Anonymous No.14669421]

>>14668151
You only proved one way, not a bijection.

[Anonymous No.14669436]

>>14669199
the contradiction is thinking that “every” infinite sequence including “some” you have no way of writing down is a reals number

[Anonymous No.14669437]

>>14669401
It can't be if it's different from each number in the list by at least one digit.

[Anonymous No.14669440]

>>14669341
>There is no way to do that.
Axiom of Choice. But you already knew that when you crafted your bad-faith argument.

[Anonymous No.14669444]

>>14668151
>Pick a random real number
Lol, how?

[Anonymous No.14669447]

>>14669440
The AC has no bearing on the process proposed by OP.

[Anonymous No.14669451]

>>14669436
Yes? This is the definition of the reals. You could express any infinite decimal as a cauchy sequence because it's obviously bounded and monotone, whether it's computable or not is irrelevant.

[Anonymous No.14669460]

>>14669440
>Axiom of Choice.
All the Axiom of Choice says in your context is that if there's a set of sets such that each subset contains one real number, you could arbitrarily choose (the one and only) element of each subset and get back R, but the crucial thing is that it doesn't imply an infinite process of countably many steps where you pick real numbers one by one.

[Anonymous No.14669472]

>>14669460
Nta, you're saying that you could partition R into singletons, but you'd need an uncountable number of steps?

[Anonymous No.14669479]

>>14669472
No, what I'm saying that you're thinking about it the wrong way when you reason in terms of constructing or partitioning things step by step. The way to think of it as if all the possible sets already exist and have always existed and you're just talking about mappings between them. "Pick an arbitrary random real number" is not a thing.

[Anonymous No.14669522]

>>14669416
>They attempt successfully to construct a new number that can't belong to this "complete" list
What do you mean can't belong to the list? It is already in the list at an arbiitrary nth position

[Anonymous No.14669530]

>>14669522
It can't be in the list if it's different from every number in the list.

[Anonymous No.14669546]

>>14669416
>the number already contains all the real numbers
>attempt to construct a new one
>reach a contradiction
Once again: if you reach a contractradiction, it means something is wrong about your reasoning. In this case the idea that you can construct a "new" real number when the list is already complete.

[Anonymous No.14669548]

>>14669522
how is this >>14669530 so difficult to understand?

[Anonymous No.14669551]

>>14669546
https://en.m.wikipedia.org/wiki/Proof_by_contradiction

[Anonymous No.14669562]

>>14669530
>It can't be in the list if it's different from every number in the list.
Who the fuck said it's different from every number in the list? Like I said, any number you can imagine is already there
>>14669369
>Every time you add a digit to your "new" number imagine the number you currently just glows at some position of the list. This continuous endlessly as the premise is the list is infinite.

[Anonymous No.14669565]

>>14669551
https://en.wikipedia.org/wiki/Principle_of_explosion

[Anonymous No.14669568]

>>14669562
So your imagination failed you. Happens to everyone every day.

[Anonymous No.14669576]

>>14669565
>can't into basic logic

[Anonymous No.14669590]

>ITT: two retards put forward mutually-exclusive propositions
>each one thinks they've proven the other wrong because theirs is incompatible with the other one's

[Anonymous No.14669595]

>>14669568
>your imagination failed you
How is my imagination failling me when the infinitists can't even imagine the number is already in the list?

[Anonymous No.14669601]

>>14669595
You said every number you can imagine, but it didn't work out that way, did it.

[Anonymous No.14669605]

>>14669551
>accuse your enemy of what you are doing, as you are doing it
I know it's a pipe dream, but I really wish mods would crack down on these woke/leftist/communist scumbags so we could have a civil conversation for once

[Anonymous No.14669608]

>>14669590
>>14669605
bots

[Anonymous No.14669610]

>>14669605
The accusation is that you clearly don't understand what a proof by contradiction is and keep going on about ex falso

[Anonymous No.14669687]

>>14669608
60 year old mentally deficient twitter user who earnestly believes people who gainsay him must be some kind of commie spy

[Anonymous No.14669830]

>>14668196
Quietly Eating Dick.
It's how every truthful OP signs off their OP.

[Anonymous No.14669877]

I don't get the logic of the diagonalization argument. It only makes sense if you say
>natural numbers are finite
>irrational numbers can be infinite
>therefore if you have a rational number for every natural number, that still leaves the irrational numbers that aren't finite
>so there are more irrational numbers than rational numbers

[Anonymous No.14669968]

>>14668151
Two infinities are not equal.

[Anonymous No.14670641]

>>14669968
The real numbers can be enumerated by the p-adic numbers, both infinities are equal.

[Anonymous No.14671023]

>>14668151
You can't even complete this or measure a non-zero amount of completion for the reals between 0 and 1

[Anonymous No.14672600]

Jesus, what a bunch of nerds

[Anonymous No.14673058]

>>14668151
You can construct such a locally 1:1 mapping, sure, but then it's not globally 1:1. In your eagerness to shitpost, don't forget that one natural number can't account for two reals. You can't just say "eh, 1243322323 is big enough that I'm sure it doesn't matter if it maps to 0.983764633 or some fraction like 73739/7363".
It just does not work that way.

[Anonymous No.14676120]

Bunp