/sci/ - Science & Math

>>8217522
>an infinite, non-repeating string in some language necessarily contains all finite strings in that language

This is so idiotic

If that was true, then at some point you have chunks of a trillion numbers repeated

imagine when, in our calculations, we reach that point, and assume that we found a period for pi

>>8217522
>it doesn't contain the finite string '11'.
but you can define

00 to be 0
01 to be 0
10 to be 1
11 to be 1

and now you can find the string 11

>>8217522
>common misconceptions that lay-people tend to have about some non-elementary mathematical concepts
0.999 = 1

>>8217530
Fuck off. The high schoolers will take this seriously.

Milhouse made the mistake of working in the ring Z/21Z.

> "Probability 1" = "Will certainly happen"
> "Probability 0" = "Can't possibly happen"

Probability Theory is tricky because you're secretly doing Measure Theory, which can be really subtle. You have a set of possible Events and on this set you put a Probability Measure, which is a special way of assigning a number between 0 and 1 to any measurable subset of events. If you have an infinitude of events, or just a weird probability measure, then it's not abnormal to have a non-empty set of measure 0, i.e. things that have a 0% probability of occurring but which are still hypothetically possible. For example, the set of rational numbers between 0 and 1 has measure 0 according to the standard Lebesgue measure, so if you pick a random number between 0 and 1 then there's a 0% chance that it will be rational, even though there are infinitely many rational numbers in this interval.

The misconception usually manifests when people talk about Borel's Theorem aka the Infinite Monkey Theorem. This (roughly) states that if you have infinitely many monkeys on infinitely many typewriters for an infinite amount of time, then there is a 100% probability that they will write every finite string. However, it's still entirely possible that they all just type "fuck you" over and over and over again.

>>8217530
Yep, 999/1000 != 1. You nailed it.

>>8217526
Or you could just not arbitrarily change the example. That's also something you could do.

>>8217533
That's definitely the wrong ring to work over.

>>8217568
Every invite string that has a chance to contain a substring does contain thus substring. Pi is (as far as we know) random. Random means that there is after every character a equal chance for ever possible next character.
Your example isn't random it still contains every substring possible.
Sage
>>>/b/

>>8217576
And here, ladies and gents, we have just what OP was talking about.
Up next: How many /sci/entists will insist that the universe is deterministic, even though we've know it isn't for nearly a century?

>>8217576
> a mathematical constant is random
8[

Sounds like you should publish your results d4wg, seems pretty ground-breaking.

>>8217576
Check out this post about misconceptions in probability theory >>8217553

>>8217576
>I think "normal numbers" means random
Epic

>Pi is a normal number
Proof ?

>>8217525
Why would anyone assume that? Mathematics doesnt work like that, because an example is not a proof.
Even if it were actually repeating, as long as we cant proof that its repeating its meaningless.

>>8217619
and in fact we can prove that it's NOT repeating, because it's irrational. (It's actually more, it's Transcendental, but proving irrationality or transcendence of pi are not incredibly easy)

>>8217522

> video games with square world maps take place on a sphere

If the top wraps around to the bottom, and the left wraps around to the right, then you're probably looking at a Torus.

>>8217627
Still waiting for a video game with a non-orientable world map

>>8217627

totaly agree.
Sometime I wish there are video game wich take place on a projective plan.

>>8217641
>>8217642
Aren't there some arcade versions of Asteroids where you play on a disk and when you reach the boundary you come back in at the antipodal point?

>>8217576
>pi is random

jesus fucking christ...

>>8217651
wouldn't that still be a torus, only with the radius in the center of the torus being zero?

>>8217651
I've only seen Asteroids on a torus, but now I'm motivated to write one that takes place on RP^2.

>>8217687
>>8217651

Just in case you don't know, remember that you can do it whit a disk or a square.

>>8217684
nop.

>>8217522

Not specifically math, but highly related:
> A implies B
> not A
> Therefore not B

This is obviously invalid, but people still do this all of the time. I TA'd an "Intro to Mathematical Logic" course once and it hurt my feels every time someone did this because it meant that I'd failed them.

>>8217756
Also not non-elementary

>For every statement you can either find a proof or a counterexample

>Pi can only contain all infinite strings that do not repeat.
>such as 3487490123127654...

Is this true?

>>8217568
yes i could

>>8217762
Like how you can't disprove a "there exists" statement with a counter-example

>>8217764
troll

>>8217762
That would mean intuitionistic logic is the same as classical logic. Are there people out there actually believing this?

John Nash was a tragic hero who overcame his crippling autism to save America from the Cold War.

>[math]\sum{n \,=\, 0}^\infty n \,=\, -\frac{1}{12}[/math]

>>8217764
It's true that's pi's decimal expansion is non-repeating.

What do you mean precisely by "contain"?

So does dy/dx mean that dy is divided by dx?

>>8217891
I like how I just learned about Leibniz notation last week and can now laugh at this.

>>8217522
"almost all"**

>>8217522
Most don't understand that something can be undecidable.

>>8217522
How exactly do you define "non-repeating"? I'd define it as not being able to find an identical sequence of digits in the string from the beginning up till the point where you find the repeat, but in that case your string clearly repeats itself in the first four digits (1010).

>>8217908
almost all what? almost all strings? no, the string
1010010001000010000010000001...
contains almost no strings in {0,...,9}*

What I'm getting out of this thread is drugs are bad.

>>8218029
Spoken like a true layperson.

>>8217522
>Counter-example: The binary string
>'1010010001000010000010000001......'
>is infinite and non-repeating, but it doesn't contain the finite string '11'.
but it's not random.

>>8217762
Ooh I know this one: Goerdel's Completeness Theorem.

:^)

>>8218044
As already said: there is no proof that pi is random or not.

>>8217989
A sequence is "repeating" if after a certain point it is just a finite string repeated indefinitely.

A sequence is "non-repeating" if it is not repeating.

>>8218044
OP didn't say a single god-damn thing about randomness.

Also, check out this post about misconceptions in probability theory: >>8217553

>>8218055
>>8218044
As already said, the statement "pi is random" literally doesn't make sense. It's not just false: mathematical constants aren't even in the realm of things to which the adjective "random" applies.

>>8217773
He didn't have autism, he had schizophrenia.

>>8217773

>John Nash was a tragic hero who overcame his crippling autism to save America from the Cold War.

he was however a schizo who saved us from the CIA

>>8217908
>>8218025
It's possible that they meant "almost all infinite sequences are disjunctive".
(a sequence is a language is called "disjunctive" if it contains every finite string in that language; let's say a number is "disjunctive" if its decimal expansion is a disjunctive sequence)

This statement is actually true, (once you clarify what "almost all" means in the sense of measure theory and specify which measure is being used on the set of all infinite sequences), and it's the correct caveat to make to OP's misconception. The problem is that most people don't realize that this caveat is needed, because they think "Probability 1" means "in all cases" and not just "in almost all cases".

With regards to pi, you can't use a probabilistic approach to the question of whether it's disjunctive or not, because pi is A SPECIFIC NUMBER. That's why we're able to compute its decimal expansion in the first place. The decimal expansion is not a random variable, though it MAY appear to BEHAVE like one. pi could fall inside this Probability 1 subset of disjunctive numbers, or it could be in the non-empty Probability 0 compliment. We literally do not know.

All we can say with certainty at this point is "pi is probably dusjunctive, maybe?"

>>8217553
>100% probability

0, actually

>>8218203

1, actually. This is a result that someone proved and you can look up.

https://en.wikipedia.org/wiki/Infinite_monkey_theorem

>>8218228
computer model simulation proves it's 0

>>8218233
gr8 b8 m8.

>>8218233
Where can I borrow infinite computing power? I could really use that right now.

>>8218236
•All the sonnets are the same length. They're by definition fourteen lines long. I picked the one I knew the opening line for, "Shall I compare thee to a summer's day?" I counted the number of letters; there are 488 letters in the sonnet. What's the likelihood of hammering away and getting 488 letters in exact sequence as in "Shall I campare thee to a summer's day? What you end up with is 26 multiplied by itself 488 times - or 26 to the 488th power. Or, in other words, in base 10,10 to the 690th.

•Now the number of particles in the universe - not grains of sand, I'm talking about protons, electrons, and neutrons - is 10 to the 80th . Ten to the 80th is 1 with 80 zeros after it. Ten to 690th is 1 with 690 zeros after it. There are not enough particles in the universe to write down the trials; you'd be off by a factor of 10 to the 600th.

•If you took the entire universe and converted it to computer chips - forget the monkeys - each one weighing a millionth of a gram and had each computer chip able to spin out 288 trials at, say, a million times a second; if you turn the entire universe into these microcomputer chips and these chips were spinning a million times a second (producing) random letters, the number of trials you would get since the beginning of time would be 10 to the 90th trials. It would be off again by a factor of 10 to the 600th. You will never get a sonnet by chance. The universe would have to be 10 to the 600th time larger. Yet the world just thinks monkeys can do it every time.

>>8218240
>Yet the world just thinks monkeys can do it every time.

1) These aren't monkeys. These are INFINITE monkeys. You're maybe not appreciating just how much bigger infinite things are than finite things. Like, by definition, bigger than any finite thing.
2) Ok, again: "Probability 1" doesn't mean "Every time."
>>8217553

>>8218240
>Now the number of particles in the universe
Mathematics doesn't concern itself with the physical universe.
>Yet the world just thinks monkeys can do it every time.
It's just a metaphor you autist. "infinite random string of letters theorem" wouldn't sound as catchy.

>>8217590
>the universe isn't deterministic

>>8218228
>>8218233

>there exists a rigorous mathematical proof of a statement concerning an infinite probability space
>but checking finitely many things is more valid

>>8218300
This wouldn't be a math thread unless somebody was complaining about a metaphor used to describe math they don't understand.

>>8217891
More or less, but not exactly.

>>8217553
>then there is a 100% probability that they will write every finite string
isnt it that the limit of the probability is 1 at infinity ?. if what arey are writing is truly random (equally likely to type any character) that is .
its the same with le wavefunction, the entire universe is located on my dick , just with pretty low probability .

>>8217522
its correct if you make the assumption that the base the digits of pi are truly random , that is all digits are equally likely when calculating the next one , which i think is true for the known subset of the digital representation of pi.
but this dosnt make sense since pi isnt 'random' its determined by the geometry of the universe .

>>8218324
it has been experimentally proven u kek

>>8217522
cant decide if the mistake is that "factor" should be "multiple"
or if 28 and 7 need to be switched

>linguistic-world problems

>>8218502

>experimentally

We are talking about Maths

>>8217522
Is confusing factor and multiple a real problem for people?

I have the most basic of white American educations and knew nothing about math, but I wouldn't do that.

>>8218597
>Is confusing factor and multiple a real problem for people?
youre mistaken
people just mix 28 and 7 up alot

I think it's genetic, you must not have the gene

>>8218511
>>8218611

>>8218497

>isnt it that the limit of the probability is 1 at infinity ?
Yes.
As the length of a random string approaches infinity, the probability that every finite string appears within it approaches 1. But, also yes, it's just a probabilistic statement and it's entirely possible that if you pick an infinite string at random it just says "fuck you" infinitely many times: if it's truly random then ANY possibility should be plausible. If you say something has "100% probability" that doesn't mean it definitely happens, it just means it "almost always" happens.

> its correct if you make the assumption that the base the digits of pi are truly random
you can also prove everything if you assume x is not x

>but this dosnt make sense since pi isnt 'random'
Absolutely correct. It doesn't make sense to talk about a constant as if it's random.

By the digits of pi being "random" I think what you really mean is whether pi is a "normal" number or not, i.e. the digits show up in a roughly uniform distribution. Here's a small thing that talks about normality, and about whether pi in particular is maybe normal: http://pi314.at/math/normal.html

>>8217525
That happened. About 200 (I think) digits in, we got the string ... 99999

And it was thought that it was just 9s from there, so they stopped calculating.

This was before modern understanding of analysis, obviously.

>>8218511
>>8218597
>>8218611
I dunno, I just googled "math misconceptions" and tried to find the catchiest photo, and one which depicted horror at a concept so rudimentary just seemed the most appropriate.

Classic Mistake, Milhouse. Classic Mistake.

>>8217590
>believing [math]i \hbar \frac{\partial \Psi}{\partial t} = \hat H \Psi [/math] isn't deterministic.

>>8218866
>believing that the wave form of a quantum system determines observations.

>>8217858
Who the fuck believes this?

>>8217756
This is the most common fallacy to exist probably.

>>8217756
>>8218961
If you commit this fallacy in real life, it's often not a big deal. If you commit this fallacy in math, you should probably kill yourself.

>>8217530
You're using irony?

>>8218819
The definition of a normal number simply corresponds to the frequentist interpretation of probability. While you are right in that every constant is either nornal or not, your BELIEF that it is random can be equipped with a probability measure under the Bayesian interpretation. In particular, if I have zero background knowledge to decide either for or against the proposition, then there is literally nothing wrong with me saying that the probability is 50-50.
This is because the probability is a measure of my belief in the proposition and has mothing to do with the proposition itself. Mathematically, a probability is nothing but a function from a collection of sets to [0,1], and it 'exists' insofar as the ground set is well-specified.

>>8217522
>The binary string
>'1010010001000010000010000001......'
>is infinite and non-repeating, but it doesn't contain the finite string '11'.
Did you check all of it, smarty-pants? Maybe it does if you go all the way to the end.

>>>Your Honor, this man claimed UNDER OATH to be certain that a sequence of digits didn't contain a certain substring, but in truth he didn't in fact check all of the numbers all the way to the end himself.
>>Is this true?
>Yes, Your Honor, but...
>>Silence! Order in the court! You are hereby found in contempt of basic common sense! I sentence you to pound-you-in-the-ass prison! Guard, take this man out of my sight! We are done with his testimony!

Hope it was worth it, Smartso the Brainiacal Wizard.

>>8219258
He's completely correct. You need at least seven nines after the decimal point for it to be equal to one.

The whole "Infinite random input mapped to alphabetical output has every string of words possible" is just a fancy way of saying that an abecedary can be arranged to say everything in our language.

>>8219343
You have absolutely no Idea what you are talking about OP VERY clearly proved that a non repeating string of 1s and 0s exists that does not contain any substring.

If you cant grasp that simple concept you should really leave this board because you clearly have no Idea about maths even on a high school level.

>>8219355
>at least seven nines
such precision is not possible in the physical universe

>>8218906
String Theory

>>8219419
but you can define

00 to be 0
01 to be 0
10 to be 1
11 to be 1

and now you can find the string 11 :)

the pattern is there

>>8219460
1=1
define 1=2
1=/=1
therefore maths is wrong

That is the shit you are doing.Of course you can redefine shit but you end up with a different string of characters or a straight up contradiction.

You are a dumb shit m8

If you have an axiom that says 1=2, then 1=2, and therefore 1=1 is still true

>>8219467
You are a dumb shit m8

If you have an axiom that says 1=2, then 1=2, and therefore 1=1 is still true

>>8219485
are you retarded?
In mathematics there is a concept of something being "well defined" meaning that it is not a contradiction and is consistent with the rest of mathematics.

Saying "define 1=2" is ill defined meaning that it is inconsistent and therefore not valid.

But what you did was even dumber you just redefined things to get a DIFFERENT series. Which makes absolutely no sense because OP without a sliver of doubt proved that a non repeating series exists that does not contain every possible sub series.
What you did was to prove that a series exists which contains the string 11.

>>8219485
>and therefore 1=1 is still true
Assuming x=x is an axiom.

>>8219490
>he thinks that math is anything but the logical conclusions that derive from a set of axioms
>he thinks you can't add a new axiom and derive more conclusions without it being contradictory

>he thinks you can't define two categories, one that includes a pair of elements, another that includes another pair of elements
>he doesn't know that patterns can include subpatterns, and all subpatterns are part of the pattern

brainlets these days

>>8219498
>he thinks that math is anything but the logical conclusions that derive from a set of axioms
I dont
>he thinks you can't add a new axiom and derive more conclusions without it being contradictory
i dont

Lets try this again:
You have a set of axioms from them you derive something. Then you define something to be a contradiction to these axioms and what you have derived from them.
That is called "ill defined".

>he thinks you can't define two categories, one that includes a pair of elements, another that includes another pair of elements
I dont
>he doesn't know that patterns can include subpatterns, and all subpatterns are part of the pattern
I dont
Have you still not understood that defining a new series means that it is DIFFERENT I though that was basic common sense.
OP proved that a series exists that does not contain the sub series "11" you proved by construction that a series that does.

>>8218824
[citation needed]

>>8219510
not him but
http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
ctrl f for 999999
or here
https://www.angio.net/pi/digits.html
even 99999999 is included in pi

>>8217522
Does the infinite string of Pi contain every possible integer represented as a finite string of digits.

>>8219522
No one knows. And proving it seems not very easy.

>>8219460
Redefine the string to just say "fuck you" infinitely many times. Now there is a pattern.

>>8219467
>>8219485
You forgot to mention that 2 =/= 1 by the Axiom of Extensionality (the one which says two sets are equal if and only if they have the same elements).

However, adding the assumption "1=2" doesn't mean "maths is wrong". You can always add axioms to formal systems to produce new formal systems. The point is that you've produced a NEW formal system with new rules, and this new formal system may not be consistent anymore, i.e. you might be able to derive a contradiction. If you can derive a contradiction then you can actually prove EVERY statement (true AND false) so the resulting formal system is utterly useless, and the added assumption was a v bad assumption indeed.

The takeaway is that is you want to prove A=>B, and you've proven (A and C)=>B, then you haven't really accomplished what you set out to do. If you've ACTUALLY proven that (A and B)=>B then you might consider killing yourself.

>>8219493
x=x actually follows from the Axiom of Extensionality

>>8219522
It's been mentioned several times in this thread that this is an open problem. Here's the google search of your post:
https://www.google.de/search?q=Does+the+infinite+string+of+Pi+contain+every+possible+integer+represented+as+a+finite+string+of+digits.&rlz=1C5CHFA_enDE504DE504&oq=Does+the+infinite+string+of+Pi+contain+every+possible+integer+represented+as+a+finite+string+of+digits.&aqs=chrome..69i57.843j0j7&sourceid=chrome&ie=UTF-8

>>8219527
>You forgot to mention that 2 =/= 1 by the Axiom of Extensionality (the one which says two sets are equal if and only if they have the same elements).
Yes exactly and as I said that means saying "define 1=2" is an ill definition because it is an contradiction to axioms or conclusions from theses axioms.

>>8219343
>you need to check every thing one at a time in order to prove a universal statement

In order to be convinced by OP's example, you need to believe in the principle of mathematical induction, or that you can construct something infinite by doing a recursive process for an infinite amount of time.

If you do, then it's easy to write pseudo-code which outputs the given string, and by design would never print two 1's in a row (I'll write one down if you want, but I have to go atm)

If you don't believe in mathematical induction (aka the Axiom of Infinity which is contained in the basic ZF package and is one of the basic building blocks of mathematics), then I guess you don't believe there are infinitely many numbers.

>>8217522
>>8219343
>>8219541
Ok, I thought of a good NON-recursive way of rigorously producing this infinite string. (although I'm secretly using the Axiom of Infinity because I'm talking about the set of natural numbers)

First: what do we mean by "infinite binary string"? What we really mean is a function
[math]f\colon\mathbb{N}\to\{0,1\} [/math]
The value of f(n) corresponds to the value of the string in the n-th place. So let's construct a function which outputs the desired binary string.

For a natural number [math]i\in\mathbb{N}[/math], let [math]\Sigma(i)[/math] be the sum of numbers from 0 to i; such sums are called "triangular numbers". Let
[math]T=\{\Sigma(i)\ |\ i\in\mathbb{N}\}[/math]
be the set of all triangular numbers.

Now let f(n)=1 if n is in T, and 0 otherwise. That is, f is the "characteristic function" for the set of triangular numbers. The first 29 outputs of f are the string in OP's post BUT with an extra 1 at the beginning: this isn't strictly a problem, but people are being pedantic about the '11' string so let's define the function g by g(n)=f(n+1), which has the effect of deleting the first character and shifting the rest one space to the left. Now the first 28 values of g(n) are exactly the string given by OP.

(proof of properties in next post)

>>8217522 (OP)
>>8219343
>>8219541

(cont. from >>8219627)


Now we need to rigorously prove that the sequence given by g(n) is non-repeating and it doesn't contain the string '11' (could also just as easily show that the sequence given by f(n) doesn't contain the string '111', you just have to find SOME finite string that it doesn't contain). Note that the difference between two successive triangular numbers is
[math]\Sigma(i+1)-\Sigma(i)=i+1[/math]
which is not constant, and grows with i. This difference measures the number of places in the sequence between successive 1's.
If the sequence given by g(n) eventually started to repeat, then the difference between successive triangular numbers would have to eventually start repeating, which contradicts the fact that the differences are strictly increasing: therefore this sequence is non-repeating.
If '11' is in the string, that means there is some number i so that [math]\Sigma(i+1)-\Sigma(i)=1[/math]. This only happens when [math]i=0=\Sigma(0)[/math]. But since g(n)=f(n+1) and 0 is not the successor of any natural number, this substring will not appear in the list of outputs for g(n)

>>8219629

Shit, fucked up the Latex, I'm sorry. Hopefully this time it works:

Now we need to rigorously prove that the sequence given by g(n) is non-repeating and it doesn't contain the string '11' (could also just as easily show that the sequence given by f(n) doesn't contain the string '111', you just have to find SOME finite string that it doesn't contain). Note that the difference between two successive triangular numbers is
[math]\Sigma(i+1)−\Sigma(i)=i+1[/math]
which is not constant, and grows with i. This difference measures the number of places in the sequence between successive 1′s. If the sequence given by g(n) eventually started to repeat, then the difference between successive triangular numbers would have to eventually start repeating, which contradicts the fact that the differences are strictly increasing: therefore this sequence is non−repeating. If '11′ is in the string, that means there is some number i so that [math]\Sigma(i+1)−\Sigma(i)=1[/math]. This only happens when [math]i=0=\Sigma(0)[/math]. But since g(n)=f(n+1) and 0 is not the successor of any natural number, this substring will not appear in the list of outputs for g(n)

>>8219629
>>8219631
Fuck this shit. Third try, no math tags:

Now we need to rigorously prove that the sequence given by g(n) is non-repeating and it doesn't contain the string '11' (could also just as easily show that the sequence given by f(n) doesn't contain the string '111', you just have to find SOME finite string that it doesn't contain). Note that the difference between two successive triangular numbers is
Σ(i+1)−Σ(i)=i+1
which is not constant, and grows with i. This difference measures the number of places in the sequence between successive 1's.
If the sequence given by g(n) eventually started to repeat, then the difference between successive triangular numbers would have to eventually start repeating, which contradicts the fact that the differences are strictly increasing: therefore this sequence is non-repeating.
If '11' is in the string, that means there is some number i so that Σ(i+1)−Σ(i)=1. This only happens when i=0= Σ(0). But since g(n)=f(n+1) and 0 is not the successor of any natural number, this substring will not appear in the list of outputs for g(n)

>>8217522
maybe you aren't grasping the concept of infinity

>>8217526
>>8219460

"the specific example X has property P"
> but if you change X in such a way that it doesn't have property P anymore, then it doesn't have property P anymore

>>8219635
Did you know that different infinite sets can actually have measurably different sizes? And that the collection of all sizes of infinite sets is so huge that they can't even form a set? And that you have to extend standard set theory with "classes" in order to even talk about "the collection of all infinities"?

If not, then it's entirely possible that maybe it's YOU who isn't grasping the concept of infinity.

>>8217756
this one cuts itself, right?
does that mean if you want a nice surface which doesn't intervene with itself for that problem you just go 4d?

>>8217891
Well, the differential is the slope of the tangent at that point, so yes

>>8219647
I think you quoted the wrong post, maybe you meant this one: >>8217722

And more or less, yes. In general, the Whitney Embedding Theorem says that if you have a smooth manifold of dimension n>0, then you can smoothly embed it (i.e. without self-intersections) into R^2n. In particular every 2-manifold (like the projective plane) can be embedded into R^4

>>8217522

Closely related to a bunch of stuff already mentioned, but worth having its own thing (kind of long, but bear with me):
> Infinity = Everything
> There's only one Infinity

The notion that an infinite set must contain everything is absurd: the set of all even numbers is infinite, but contains no odd numbers. Nevertheless, I think people are implicitly making this mistake when they make claims like in OP's post, and like >>8219635.

As for "different Infinities" you need to specify what is meant by two infinite sets having the "same size". In order to compare two abstract sets, the only tools you have are functions between them.
Say that a function f:A->B is an "injection" if whenever f(x)=f(y) then x=y; in other words, f is embedding A as a subset of B, so we could define A to be "at most as large as" B if there is an injection from A to B.
Say that a function f:A->B is a "surjection" if for every y in B there is an x in A so that f(x)=y. Since there are no elements of B which are not hit by this function, it's safe to say "A is at least as large as B" if there is a surjection from A to B.
Say f:A->B is a "bijection" if it as an injection and a surjection. A bijection is a 1-to-1 correspondence between elements of A and elements of B, so in light of a perfect matching such as this we might say A and B have "the same size" if there is a bijection between them. (there's lots of good, simple theory lurking here, please look it up)

Say a set is "finite" if there is a bijection between it and some bounded subset of the natural numbers; say that a set is "infinite" if it is not finite.

(continued in next post)

>>8219786

Claim: the set of natural numbers N and the set of integers Z have the same size. You can recursively construct a bijection by "counting" Z:
0, 1, -1, 2, -2, 3, -3, 4, -4, ...
(if you're not satisfied with "..." I encourage you to write down an explicit function N->Z along these lines, by splitting up into odd and even cases. If you do it right, every integer should be hit precisely once)

Call a set "countable" if it has the same size as N.

Claim: the set of rational numbers Q is countable. This is harder, but the intuitive idea is that Q is the same size as the set of all pairs of integers, and the set of all pairs of integers has the same size as the set of all integers, which has the same size as N. Rigorous details can be given and usually are in a first course about pure math (you'll probably need the Schroeder-Bernstein theorem).

Claim: the set of real numbers R is NOT countable, i.e. there is NO bijection with N. The proof is by Cantor's Diagonal Argument, which I won't repeat here in its entirety (https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)), but the rough idea is not so outlandish: suppose there IS a bijection f:N->R, and arrange the decimal expansions into an infinite array, so that the n-th row is the decimal expansion (after the decimal point) of the number f(n). Now construct a real number r by inductively running down the diagonal of this array and choosing a DIFFERENT digit for r's decimal expansion. If you do this in a clever way, you will have constructed a real number which is not in the image of your bijection, which is a contradiction.

(more continued)

>>8219786
>>8219790

There something much more general that can be said: for any set A, let P(A) denote the "power set", i.e. the set of all subsets of A. Then Claim: A and P(A) do not have the same size.
Suppose, for the purpose of contradiction, that there is a bijection f:A->P(A) with inverse g:P(A)->A. Define
[math]C=\{a\in A\ |\ a\not\in f(a)\}[/math]
This is a perfectly fine subset of A, i.e. an element of P(A). Now we can ask: "Is g(C) an element of C?"
If yes, then by definition g(C) is not an element of fg(C)=C, which contradicts the "yes"
If no, then by definition g(C) IS an element of C, which contradicts the "no"
In either case we have a contradiction, so the assumption that there was a bijection must be inconsistent.

So you could form a sequence
A, P(A), P(P(A)), P(P(P(A))), ....
of sets whose sizes are strictly increasing. But no many how big A was, and no matter how many times to apply the power set operations, these sets are still tiny because you could take the union of all of them and then form the power set of THAT. This is the theory of Cardinal Numbers (https://en.wikipedia.org/wiki/Cardinal_number)) and it's a really good place to start if you want to actually understand how infinite stuff works. Every Pure Math student is expected to be at least acquainted with these notions.

>an infinite, non-repeating string in some language necessarily contains all finite strings in that language
>non repeating string

Your counter-examples include repeating strings, that is why you can for say for sure that 11 does not exist. in that sequence. While non-repeating is different.

>>8219807
By "non-repeating" I mean it doesn't eventually become an infinite repetition of a fixed finite string. What do YOU mean by non-repeating?

>>8217522
ITT:
- people who already know what they're talking about and who don't hold these misconceptions
- people who swoop in, spout uninformed nonsense based on vague intuition and non-standard definitions, and then disappear forever

I'm not sure if anyone is actually benefiting from this thread.

>>8217525
Pi has been proven to be irrational, hence no period

>>8219629
>>8219627
>>8219541
>>8219419
Guess I found the autistic thread.

>>8220048
ur dum

>>8217891
>>8217899

You joke, but Robinson wrote a book in the 60's which constructed a non-standard model of real analysis, where the real line also contains "infinitesimal quantities" which are smaller than any positive number but bigger than 0. Arithmetic extends to include the infinitesimals and the definition of the derivative doesn't need to use limits anymore, and COULD be interpreted as the ratio of two infinitesimals.
https://en.wikipedia.org/wiki/Non-standard_calculus#Definition_of_derivative

>>8218906
Memelogists

>>8217522
PEMDAS

but more specifically "left to right"

>>8220348
solve this differential equation
[math]dy \div dx = y[/math]

>>8220511
So many "if u solve dis ur genius" memes on facebook that are just testing if you know the order of operations, and so many people fucking them up.

>>8220522
y=e^x, or at least approximately?

>>8220522
[eqn]
\frac{dy}{dx} = y
[/eqn]
Let [math]y = e^x[/math]
[eqn]
\begin{aligned}
y &= e^x \\
y' &= e^x \\
\ldots \\
\frac{dy}{dx} = y
[/eqn]
[math]e[/math] is your best friend in these simple examples of DEs since the exponential function is literally defined as its own derivative. If you hate e, you'll hate everything DE.

>>8219520
Yes, obviously the 9s are actually there. A citation is needed on the idea that people believed it was repeating from that point, because it sounds like total bullshit.

>>8217522

>The 4th dimension is time

The (n+1)-st dimension is whatever the fuck you want it to be, as long as it isn't a linear combination of the first n.
https://en.wikipedia.org/wiki/Vector_space#Basis_and_dimension

In the classical model of space-time, which has three spatial dimensions and one temporal dimension, I guess you could say that the time dimension is the "fourth" one.

>>8220534
The thing is there is no order

maybe it has become so common that we can call multiplication and division before addition/subtraction etc a commonly accepted convention,

but when you have ambiguity between multiplication and division there is no correct order it is ambigous

but autists will still valiantly claim that their order (left to right) or other disambiguating rule (say, implied multiplication) is the right one

>>8218906
who doesnt

>>8220522
The family of solutions is

[math] y=e^{x-c} [/math]

Where c is the constant given by:

[math] \int \frac{1}{y} \cdot dy [/math]

>>8217762
Why isn't this true?

>>8217522
Couldn't you say "irrational number" instead of infinite and non-repeating. That would exclude patterned numbers.

>>8218240

t. layman

>>8221361
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

>>8221422
is it true if you allow arbitrarily picking the axiomatic system?

>>8221495
a statement only makes sense inside an axiomatic system

>>8217525
>chunks of a trillion numbers

>>8217522
>this
>common misconception

What? Just look at the Fibonacci numbers. It was literally the first thing I learnt in algebra class.

>>8221313
>PEDMAS
>DM
>ambiguity

>>8221363
A real number is irrational iff its decimal expansion is not eventually periodic, i.e. what I'm meaning by "non-repeating"

What's a "patterened number"?

>>8221495
If you pick inconsistent axioms then everything is true.

>>8221755
4/5*3

ambiguous
it has no "answer"

>>8221767
...assuming the principle of explosion

t. paraconsistent logics

>>8221535
What am I supposed to be seeing by looking at the Fibonacci numbers? It's a sequence of natural numbers whose successive ratios converge to the golden ratio. Cool?

The first thing I 'learnt' in analysis class is that math doesn't always work the way your intuition says it should.

>>8221779
0 to the power of infinity rubs me the wrong way

>>8221782
who cares if you get a rub

>>8221782
Show us on the extended real line where [math]0^\infty[/math] touched you.

>>8221770
That's not ambiguous. PEDMAS, guy. It's 12/5.

In situations like this, there needs to be a convention. And that's why there is, it's called PEDMAS. Evaluate the division symbols first, and then the multiplication symbols.

If you really want to make sure stuff's unambiguous, add parentheses.

>>8221788
Who on earth even writes fractions with '/' outside of computer programs?

>>8221788
>Evaluate the division symbols first, and then the multiplication symbols.
that is not an established convention

maybe, MAYBE some calculators and programming environments have chosen a convention for disambiguating it, but in real life it is not a well-formed expression

even 1+2*3 is ambigous strictly speaking

>>8221789
People who do math.

>>8217525
>>8218240

> a large finite number is a good approximation for infinity
From the perspective of an infinite set, there are no "large" finite sets. A countably-infinite set (the smallest size of infinite sets) CAN be exhausted by finite sets but you still need infinitely many of them.

>>8217522
This one is more opinion-based than factual:

> Banach-Tarski is a "paradox" that couldn't possibly happen in the real world, so the Axiom of Choice "ought to be" rejected
> formalism should reflect our perceived reality

The Axiom of Choice is logically independent from the Zermelo-Fraenkel axioms so, aptly, you have a choice over whether you assume it or not. If you assume it's true, then you can prove a whole bunch of cool stuff; if you assume it's false, then some of the basic notions of Quantum Mechanics don't make sense anymore because not every vector space would have a basis. There are probably good non-AC models of physics, but QM works well enough that I'd rather not abandon it over vague philosophical irks.

Ultimately, formalism has no moral obligation to adhere to how people think reality should work, and in the past it has been very successful at shattering perceptions (see: the practical applications of Relativity Theory). Just remember: numbers aren't real.

>>8221789
Russians

>>8217522
>math is cool
Wrong.

>>8223166
Sad but true

>>8221788
Its a bit ambiguous
I've read texts where they use a/bc=a/(bc), to avoid parentheses.
They figure if you meant (a/b)c, you'd have written (ac)/b.

>>8221785
Suddenly it makes sense why Wildberger hates real numbers

>>8217522
if the digits were random then would it be true

>>8223166
>>8223244
I love math, and I currently do it for a living, but I think it's about 2% cool. 98% of the time it's just tedious and boring.

>>8223166
FACT: math is not a physical object and does not have a temperature.

>>8223702
they are random

otherwise it would have a pattern

>>8223702
Not necessarily. What is true is that if you pick an infinite string via an random process, then there is 100% probability that it will contain every finite string (this is colloquially referred to as the Infinite Monkeys Theorem). BUT, 100% probability only means "almost always" and not "in every event," so it's still feasible that you could randomly pick a constant string. See >>8217553

>>8223953
What are random? Are you talking about the string in OP's post? Because there's definitely a pattern, it's just not periodic

>>8223953
PI is not random.
There is an explicit formula to calculate the nth number of pi.
How can something like that be random?

>>8224131
>There is an explicit formula to calculate the nth number of pi.
that's just an approximation, not the real pi

>>8224133
firstly you are wrong https://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_formula

secondly that doesnt change anything.
Pi has a pattern it is a very complex one but because there is a formula to perfectly calculate it up to a given digit by definition it cant be random.

1-1+1... = 0.5

>>8224156
Biggest bullshit ever

Does String Theory really depend on this?

>>8224133
In practice you can only ever compute finitely many digits of pi (i.e. an approximation) but the digits are still deterministic. That's why we're able to compute them at all.

It doesn't make sense to call a constant "random".

>>8224168
no

What is used in the meme case, is the
fact that on it way to infinity, the smoothed asymptote hits the point (0, -1/12)

Think of it as a kind of tornado, the top hits infinity of course but the tail at x=0
touches y= -1/12.

Different non-convergent series go to infinity, but the route there varies, and the series can be classified based of that.

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#/media/File:Sum1234Summary.svg

>>8217522

>I don't personally know a specific value of a something, therefore it is random.
>Constants can have random properties.
>"arbitrary" = "random"

>My vague conception of "randomness" agrees with the definitions of Probability Theory

>>8224168
>I don't understand something
>but that thing sure is bullshit

>>8217522
>analytic geometry = algebraic geometry = arithmetic geometry
sometimes it's hard to tell if someone's confusing these or not, but other times it's like "holy shit you seriously think that algebraic geometry was invented by Ronnie Deckhart."

>>8224156
is that 1 + (-1) + 1 + (-1) ... ?

>>8226487
yes

>>8224156
>computing the limit of a divergent sequence

>>8221788
but it's PEMDAS not PEDMAS, you fucked up

> dy / dx = y/x

>>8218890
>what are eigenvalues

>>8220552
>>8220557
>>8221352

>believing [math]dy \div dx[/math] is a derivative and not a fraction.

The misconception that [math]\displaystyle \sum_{n=1}^\infty n = - \frac{1}{12} [/math].

The misconception that [math]\displaystyle \sum_{n=1}^\infty n = - \frac{1}{12} [/math].

The correct statement is [math]\displaystyle \sum_{n=1}^\mathfrak{R} n = - \frac{1}{12} [/math].

The misconception that [math]\displaystyle \sum_{n=1}^\infty n = - \frac{1}{12} [/math].

The correct statement is [math]\displaystyle \sum_{n\geq 1}^\mathfrak{R} n = - \frac{1}{12} [/math].

>>8219639
>standard set theory
NBG 4 lyfe, yo.

>>8229633
What is R in the lower sum?

>>8222469
Without Choice you can't even prove that surjective function has a right inverse, there are infinite non-Dedekind infinite sets and some fucked up ordinals appear. So I'd rather solve world starvation by infinitely copying a cheese ball than allow shit like that in MY math.

>>8229672
all real numbers?

>>8229694
That is not the real numbers notation nigga.

>>8221495
No. The statement that can neither be proven true or false, syntactically in your axiom system (the so called "Gödel sentence" G), depends on the set of axioms.
And if you take the axioms and add some other sentences as, then you get a new Gödel sentence G'.

For example, G might be the Gödel sentence for one axioms system
S1={A,B,C,D,...}.
And if you add G to the axioms
S2={G,A,B,C,D,...},
then of course G is provable from S2, but then there is another sentence G2 that's not provable from S2
(assuming consistency)

>>8229672
It's the Ramanujan sum.

>>8217722
Why would you make the jif loop immediately, rather than give time to try and comprehend the geometry of the form?