/sci/ - Science & Math

Pi or Tau?

>>11525093
Whether the RH is true or not.

>>11525093
They don't. If you disagree you're a "crank".

>>11525093
whether OP's faggotry is countably or uncountably infinite

0.9999999999999999999999... = 1

>>11525093
With and for their paychecks.

>>11525102
this. after hundreds of years there's still no definite conclusion to the debate

>>11525102
>>11525224
Yes there is, retard. The answer is yes and it follows trivially from the definitions. Stop pretending you actually know any maths.

>>11525228
for every definition that allows it, the same definition disallows it.

>>11525229
How about the standard, implicit definition that
0.d_1d_2........
is defined to be the smallest real number not smaller than any of the elements of the set
{0.d_1, 0.d_1d_2, 0.d_1d_2d_3,..} (the set of finite truncations).
1 is the smallest real number not smaller than any of the {0.9, 0.99, 0.999, ....}. You're saying that the same definition disallows 0.999...=1.
Prove how. That is, give a real number (via a representative of the equivalence class of Cauchy sequences or a dedekind cut) that is smaller than 1 but not smaller than any of the numbers 0.9, 0.99, ...
Oh what, you've never even heard of these definitions? That's because you're a retard who doesn't even know what he's talking about. Just shut up and stop embarrassing yourself.

>>11525240
for the set
[math]A: [0.9, 0.99, 0.999, 0.9999, 0.99999, ...][/math]
requires adding the same n'th element in
[math]B: [0.1, 0.01, 0.001, 0.0001, 0.00001, ...][/math]
to equal 1.
aka
[math]A[/math] = the partial sums of [math]\sum_{n=1}^{\infty} \frac{9}{10^n}[/math]
[math]B[/math] = 1-A for every partial sum of A

at what point do we just ban .999...=1 fags on site?

>>11525253
No the sequence A=(0.9, 0.99, 0.999, ...) is just a Cauchy sequence REPRESENTING the equivalence class of 1 in the real numbers. Remember, the equivalence relation is that 2 sequences a_n, b_n are equivalent if and only if for all e>0, there exists a natural number N such that for all n>N, |a_n - b_n|<e.
Clearly (0.9, 0.99, 0.999, ..) is equivalent to (1, 1,1,1,1,...) because the sequence (0.1, 0.01, ...) approaches zero in the sense that for rational numbers e>0, the numbers in the sequence are all eventually below e.
It's an extremely simple construction. How can it take retards like you SO LONG to understand it?

>>11525311
>>11525253
>>11525229
>>11525224
>>11525102
Mathematically illiterate retards tongue my anus.

>>11525312
>approaches 1
>approaches 0
these are fundamentally useless notions if we're talking about infinity.
0.1 is
0.100000000000000000000000000...
0.010000000000000000000000000...
0.001000000000000000000000000...
0.000100000000000000000000000...
how do you approach something that you can't get close to?

>>11525332
God dammit you're an idiot. I hope you have realized this by now. Still, I'll try and explain it to you.
The word "approach" is not some vague metaphysical word whose meaning is relegated to philosophy. It has a precise, mathematical definition.
Mathematicians say that a sequence of rational numbers (q_i) where q_i is the i'th rational number in the sequence approaches the rational number x if for every rational number e>0, there exists a natural number N such that for all natural numbers n>N, |q_i - x|<e.
Simple definition. No vague reference to infinity is involved. All we assumed is that for every natural number i we have a member of a sequence q_i.
Now with this standard, thoroughly uncontroversial (and how can it be controversial? it's just a definition) definition, the sequence (0.9, 0.99, ...) approaches 1.

>>11525228
0.999... is not equal to 1 itself, but it's limit is equal to 1, and that is a difference

how many times can you cut a piece of A4 paper in half?

this notion is not applicable to infinity.
if infinity were applicable, then you could always continue cutting smaller pieces of paper. you could cut for 12 hours, sleep, wake up and do it the next day for 12 hours, and continue doing this for weeks, months, years, because there is no smallest piece closest to 0, just as there is no largest number closest to infinity. There is no final cut, and every recent cut still produces a small amount of paper.

Surely at some point you'd get bored of cutting paper pieces smaller than planck lengths and move onto something else though.

if theres no end to the work, there's no end to the work - that's just how it be. You can't approach 0 paper cause every cut you make creates two equal halves that can be cut again.

alternatively, if you had a piece of paper with infinite length and infinite width, where does the initial first cut even begin? You could just cut from wherever you are and begin slicing through the paper, but you'd never make it from one edge to the other and never make a slice all the way that actually creates 2 halves.

>>11525342
>No vague reference to infinity is involved
>the sequence (0.9, 0.99, ...) approaches 1.
>(0.9, 0.99, ...)
>, ...)
>...
>No vague reference to infinity is involved
mind explaining what the dots mean, then?

>>11525093
There are some small things they disagree on, like which area of maths should get more funding or more research in it (Grothendieck famously held a highly controversial view of where mathematics should be headed). Mathematicians also disagree about what they believe the truth values of various unsolved conjectures to be. There are also some disagreements on how mathematics should be done, intuitionistism vs classical mathematics, as well as the usefulness and desirability of machine proof verifications. Some disagreements over the relationship mathematics should hold with regard to physics (famous V.I. Arnold's quote "Mathematics is a part of physics").
Sometimes published proofs turn out to be wrong and the mathematicians on all sides work very hard to resolve the issue. Sometimes arrogant cranks claim to have solved a problem which they haven't and refuse to honestly and humbly try to explain their approach as well as refusing to listen to arguments for why they are wrong (high tier example: Mochizuki, shit tier example: Tooker).
Despite all these agreements, I believe there is a fundamental understanding between the mathematicians that they're just looking for truth and are fundamentally honest and willing to listen to each other. One of the reasons I decided to study mathematics is because of how self-consistent and uncontroversial it is. Most mathematicians, I believe, are humble and eager to listen.

>>11525353
>>No vague reference to infinity is involved
>>the sequence (0.9, 0.99, ...) approaches 1.
>>(0.9, 0.99, ...)
>>, ...)
>>...
>>No vague reference to infinity is involved
>mind explaining what the dots mean, then?
Sure! The dots here is just me trying to abbreviate the concept. It's in no way formal notation. Dots represent a pattern that you should notice (adding 9's).
Formally, I would say that the sequence is the function f:N->Q (from natural numbers to the rational numbers) where f(n)=1-10^n. This is the precise definition of the sequence (0.9, 0.99, 0.999,...). As long as you accept that there are natural numbers, you're good.

Please do yourself a favor and read some math.
And the dots are just notation. Nothing ambiguous there.

Axiom of choice, continuum hypothesis

>>11525361
Thank you! I've been losing my mind trying to argue with these idiots.
Read Rudin before trying to argue that 0.9999...!=1, everyone!

>>11525361
what does the notation imply?
it's continued, sure.
there may (or may not be) a pattern, sure.
how long is it continued though?

this is the part where you mathlets trip.

>>11525389
The notation implies a function f:N->Q (from the natural numbers to the rationals) defined by the formula
f(n)=1-(0.9)^n
simple
>it's continued
What does that even mean?
>how long is it continued though?
It's defined on all natural numbers.

You are the one that's tripping through your retardation.
>mathlets
I'm better at math than you will ever be. I go to a world's top 5 uni.

>>11525427
i'm sure you're planning to do nothing of value with your education besides joining academia and becoming math professor yourself, of which everyone around you is also being railroaded down the same path so goodluck even getting that job. I'm sure they'll need a math teacher for some class somewhere on earth where you'll have to relocate to though.

>>11525093
Is mathematics a worthwhile endeavor?

>>11525253
exactly, [math]0.999... = \sup A[/math]
[math]\sup A = 1[/math]
you do know what "supremum" is, don't you anon

>>11525438
Way to derail the discussion.
Seethe harder mathlet.

>>11525445
1 is not an element of A you circular logic brainlet.

do you know what supremum is?

>>11525452
A supremum of a set doesn't have to be an element of the set. Holy shit you're actually retarded. You don't even know the definition of a supremum.
Here, let me help, moron.
>The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists.
https://en.wikipedia.org/wiki/Infimum_and_supremum
Learn to read.

>>11525447
being trained to talk nonsense is indistinguishable from coming upon it in hysterics. You're a well trained parrot regurgitating nonsense.

>>11525452
>t. don't know what supremum is

>>11525454
maybe you should learn to read, else whatever way you did learn to read is really throwing off your ability to understand that definition.
T = A
S is a subset of T. Aka, S is a subset of A.
Supremum is the [math]LEAST[/math] element [math]IN[/math](subset) T that is greater than or equal to all elements of S.

show me the part where 1 is in T.
Where 1 is in A
>A: [0.9, 0.99, 0.999, ->]

you're walking into the problem with the presumption that "0.999...=1" already and therefore assuming 1 exists in A.
circular logic.

>>11525464
T is the set of real numbers
S is A.

>>11525464
>Supremum is the LEASTLEAST element ININ(subset)
no, supremum is the least element in the ambient set T, not necessarily belonging to S.

A as a subset of the real numbers has a supremum which doesn' belong to A. this supremum is equal to 1.

A as a subset of itself doesn't have a supremum.

>>11525464

>>11525471
>A as a subset of itself doesn't have a supremum.
And whats the supremum of the set of all integers?
[math]C: [0,1,2,3,4,5,/dots] [/math]

>>11525474
this set doesn't have a supremum

>>11525475
C is bijected with A, you understand?

>>11525477
That's irrelevant, moron.

>>11525477
>>11525474
>>11525464
You're just embarrassing yourself at this point.

>>11525477
C as subset of C doesn't have a supremum
A as subset of A doesn't have a supremum
you understand ?

>>11525093
How best to teach and write math, surely.

>>11525481
is this the language they teach you in your courses?
is [1,2,3] actually definable as of a subset of [1,2,3]?
a subset of [1,2,3] is [1,2] or [2,3]. if you combine them, you just get the whole set, it's a not subset. keyword SUB. less. smaller.

the definition of the supremum should have no issue working here unless it's not defined well for infinite sets, which appears to be the case given the requirement of having a greatest value, of which no greatest value exists in an incrementing infinite set.

there is no supremum for A. there is a supremum for B though, and it's 0.1

>>11525498
In modern mathematical terminology, a A is called a subset of B if and only if every element of A is also an element of B. Hence every set is a subset of itself.
>the definition of the supremum should have no issue working here unless it's not defined well for infinite sets,
Supremum is defined for some infinite sets, but not all.
For example, as you mentioned, the natural numbers don't have a supremum as an ordered set, but the set [0,1] (the closed unit interval) does, and the supremum of [0,1] is 1.

The issue here is miscommunication. An ordered set has a supremum in itself if and only if it has a greatest element, that's true.
However, in the case of 0.9999...., it's defined as something a bit different. It's not defined as the greatest element of the set A={0.9, 0.99, ...} but rather as the least number of R( the real numbers) that is not smaller than all of the elements of A. There might be no element of A that is not smaller then all of the elements of A, but there might be an element outside of A but in R that is not smaller than every element of A.

A fundamental theorem about the real numbers is that for every subset X of the reals, if there is a real number that is not smaller than all of the elements of X, then there is *the unique least* element of R not smaller than every element of X.
Going back to the set A={0.9, 0.99, ...}
It has an upper bound in the reals, which is 1, because 1 is greater than every element of A.
It turns out, 1 is also the smallest upper bound of A. That means 1 is the smallest element of the reals that is not smaller than every element of A.
This is how mathematicians define 0.99... and 0.a_1a_2.... for every sequence of numbers a_i.

>>11525477
you do know that bijections don't preserve properties of boundedness right anon?

there is a misunderstanding indeed and that its the failure to realize the mapping procedure between C and A.
[math]A: [0, 0.9, 0.99, 0.999, \dots ][/math]
[math]C: [0,1,2,3,\dots][/math]
A and C are bijected. Indexing from 0, every n'th element in C is equal to the n'th partial sum in [math]\sum_{n=1}^{\infty} \frac{9}{10^n}[/math], which is also equal to the n'th element in A.

The deceitful process occurring here is probably the result of old, poorly written definitions.
Although we say A must exist in the range [0,1], we can't forget that [0,1] of A is mapped to [0,∞] in C. Even just saying that is fucking nonsense since ∞ isn't a number, but I hope i'm getting across the definition of a mappable range that is related to bijection.

What this really means is that the "1" we see here is not the actual real number 1. It's a stand-in for infinity. Just as ∞ doesn't actually exist as a "number" in C, neither does 1 exist as a "number" in A.
You can say that, logically, in finite integer counting terms that 1 is a real number in the number line, 2 comes after it, etc; but truthfully this 1 when mapped with infinite values between [0,1] is as ethereally garbage as ∞.

To say 1 is the supremum of A necessarily equates to ∞ is the supremum of C; but regardless whether both of these are true or both not, there is the logical comprehension issue of treating the "1" A appears to be approaching as the proper integer [math]1.0[/math] of which must be the least greatest number over the elements of A.
But again, this isn't the case. This isn't [math]1.0[/math], this is merely a stand-in for ∞, and since ∞ isn't even a number much less an integer, neither is this 1.

infinity is absolutely fucking retarded in math because of how it works both ways and creates ambiguity.

>>11525561
put it another way, if 0.999...=1
that means 1 is the finite end of the decimal expansion of 9's.

There is a big comprehension issue occurring here if it's allowable to say INFINITE numbers can(or must) exist BETWEEN 0 and 1. Because that necessarily means 1 comes after an infinite amount of numbers that proceed from 0.
Are we allowed to say anything can come AFTER infinity?
If we're told to believe this is all true, then what's really being said is
>[math] A_2: [0, 0.9, 0.99, 0.999, 0.9999, \dots , 1.0 ][/math]
which is not fucking different from
>[math]B_2: [1, 0.1, 0.01, 0.001, 0.0001, \dots, 0][/math]
but then putting shit AFTER infinity also directly relates to being allowed to say
>1 - 0.999...9 = 0.000...1
and if we go back to where I defined what the fuck A even was as [math]\sum_{n=1}^{\infty} \frac{9}{10^n}[/math], then we necessarily arise at the issue of where 1.0 in A2 comes from and where 0 in B2 comes from, which further necessarily requires [math]\frac{9}{10^{\infty}}[/math] to be an element, requiring infinity be not only a number that exists in the set, but also an actual integer that can be properly incremented to from the reals.

infinity is buttfucking retarded.

>>11525561
A doesn't contain 0.999... it only contains 0.99..9 for all finite number of 9s. it's in a bijection with C, the set of natural numbers (which you keep wrongly calling integers), which preserves the ordering.
A doesn't have a supremum in itself
C doesn't have a supremum in itself
A has a supremum as a subset of the real numbers.
this doesn't imply that C needs to be embedded in some larger set where it has a supremum. you're right that this supremum would be what we intuitively call ∞
there is absolutely no contradiction

>>11525601
addendum
>[math]C_2 : [0,1,2,3,\dots,\infty ][/math]

>>11525604
1/3=0.333...
1/3*3=0.333...*3=0.999...=1
prove me wrong

>>11525605
you're trying to sneakily pass the properties of infinity onto a real number. that's the problem here. that's the contradiction. infinity is explicitly not defined to work like a real number, and the usage of "infinite limit convergence" or supremum here is just trying to pass the properties of infinity onto the real number 1, but then saying that this is the real number 1 that goes to the real number 2; aka using the real number properties that are not congruent with the usage of infinity.

come on man.

>>11525609
0.999... = 1, but your argument is not formal and if you make it formal you it will become overly complicated.
First you have to:
1) prove that 1/3=0.333... . For that you have to again define what you mean by 0.3333... and infinite expressions in general. But if you define infinite expressions in general,0.999.. follows trivially so all you're doing is adding unnecessary steps.
2) Prove that 0.333...*3 = 0.999... Sure, the intuitive idea is that you multiply each digit by 3 and that's the answer. But this is not a mathematical proof. The fact that finite decimal expansions behave like this doesn't mean infinite decimal expansions should.

In conclusion, 0.999... IS equal to 1 but your proof is not good, and also not a proof of anything.

>>11525614
>you're trying to sneakily pass the properties of infinity onto a real number.
am I ?

>>11525253
>requires adding the same n'th element in
Incorrect.

>>11525616
>your argument is not formal
basic logic is formal so it is formal

>prove that 1/3=0.333
3 1s fit into 1, remainder 1, this repeats without end

>what you mean by 0.3333... and infinite expressions in general
it repeats without end

>Prove that 0.333...*3 = 0.999
3*3=9, this repeats without end

>your proof is not good
this is just an informal subjective opinion, it just doesn't "seem right" simply because it is short

whether 0 belongs to N

>>11525636
>repeats without end
>repeats without end
>repeats without end
>why?
>because it repeats without end
You're as bad as the 0.999...=/=1 morons. Honestly, just pick up a book on analysis. Is it that hard?

Infinity is too big bros.
everything that has been counted by humans since the dawn of human pre-history, added together, regardless if had been counted multiple times by different counters, added with all the numbers that ever were counted or instantiated in an equation, is a bigger real number than you can comprehend or find usefulness in, and it's still smaller than infinity.

Put down the weed. There is absolutely no need for infinity in math. It solves no problem.

>>11525643
Ok so how many natural numbers are there?

>>11525644
(undefined)

>>11525648
Claiming infinite sets don't exist implies that all sets are finite, which in turn implies every set has a well-defined finite number of elements.
So I ask again, how many elements are there in the set of natural numbers?

>>11525614
>you're trying to sneakily pass the properties of infinity onto a real number. that's the problem here. that's the contradiction. infinity is explicitly not defined to work like a real number, and the usage of "infinite limit convergence" or supremum here is just trying to pass the properties of infinity onto the real number 1
[math]5 = \sup \{ x \in \mathbb{R} \mid x < 5 \} [/math] how weird is that

>>11525652
Uh, no.
Claiming infinity doesn't exist is simple as.
It doesn't mean you must assume a greatest finite number must exist.

I'm saying stop chasing the carrot on a stick, not pretend the carrot on a stick is cucumber on a stick.
You're trying to prove the same illogical nonsense
>1. THERE MUST BE A GREATEST VALUE
>2. IF ITS NOT INFINITY THEN WHAT FINITE NUMBER IS IT

there is no greatest value. Do not pass go. Do not collect $200. Do not proceed to 2.

>>11525655
retard

>>11525660
[math]5 = \sup\{ 5 - \frac{1}{n} \mid n \in \mathbb{N} \}[/math] HOW WEIRD IS THAT

>>11525668
>>11525655
Thank you for at least trying. I have tried too but these retards are very unreasonable. They literally can't understand basic logic. You are obviously right, but some people are just unaffected by reason.

>>11525668
double retard, also not true.
also not valuable information even if it were true, which it isn't, which further proves how worthless it is since it being wrong doesn't matter to anything.

>>11525601
>put it another way, if 0.999...=1
>that means 1 is the finite end of the decimal expansion of 9's.
No, it doesn't.

>Because that necessarily means 1 comes after an infinite amount of numbers that proceed from 0.
That's wrong though, what is the number that proceeds from 0? What is the number that precedes 1?

>Are we allowed to say anything can come AFTER infinity?
It depends what you mean. If you mean by infinity an unending list, then there is no end of the list to come after. If you mean there are uncountably infinite numbers between two numbers then that is fine. But you attempted to argue as if these are the same thing, a faulty conflation.

>but then putting shit AFTER infinity also directly relates to being allowed to say
>1 - 0.999...9 = 0.000...1
Doesn't follow. Your set is not a Cauchy sequence.

>then we necessarily arise at the issue of where 1.0 in A2 comes from
A2 is irrelevant.

>>11525093
Whether the axiom of choice and law of excluded middle (either P or not P) should be accepted.

>>11525680
Crotchy diddlercut retard is back

>>11525684
Not an argument, thanks for admitting 0.999... = 1

>>11525680
>>11525687
Based mathematically literate anon trying to school gorillas.

>>11525687
crotchy diddlercut schizo retard who has make-believe conversations that agree with himself arbitrarily to serve self-aggrandizing delusions of grandeur over trivial nonfuctional knowledge that defeats his own ability to use logic correctly.

>>11525693
I'm not he anon you're responding to. He's not a samefag. You're the schizo retard.

>>11525701
You should take issue with him pretending to be you.

>>11525093
https://en.m.wikipedia.org/wiki/P_versus_NP_problem

>>11525679
>also not true
prove me wrong.
wait, you can't. I guess that means you're gonna have to watch me fly
[math]1 = \sup\{ 1 - \frac{1}{10^n} \mid n \in \mathbb{N} \} = \sup \{ \sum_{k=1}^{n}\frac{9}{10^k} \mid n\in \mathbb{N} \}[/math] HOW FUCKING WEIRD IS THAT

>>11525708
Or in other words, "in any cryptographic algorithm, like AES is there some secret backdoor algorithm that lets you break it?"

>>11525680
don't you understand how sad and fleeting your desperate grabs at legitimacy are?
Don't you get how much defending infinity makes you look like a fool?
You're asking me to name the immediately proceeding real number from 0? Preceeding 1?
In relation to your defense of there being infinite numbers between 0 and 1?
I do not believe for a second you actually comprehended the post you quoted so I can't give you any points or a reason to continue this conversation since you're clearly instantly going off on some strange tangent of chasing your own ass.

>>11525693
See >>11525687

>>11525715
See >>11525716

>>11525709
Proved it wrong already niggernogger, unless you're trying to claim there aren't infinite numbers in the set of N.

Go back and read.

You niggers are getting trolled

>>11525093
They disagree on things like "is mathematics invented or discovered", or "is the law of the excluded middle a valid one to hold universally or should you have to prove its validity on a case by case bases?" or "Does P equal NP?" or the true ultimate one: "What is the qualities of the ETERNAL GOD" and some other things of that nature.

It is so sad what academia does to people who can't get good grades in any othet subject than math.

Just fills your head with nonsense bullshit to prep you for becoming a math teacher where'll you'll either fill other people's heads with bullshit or be well qualified to teach middle school math following the government mandate curriculum.
Cauchy crotchy diddlekid cuts and infinite definitions of the magical mcguffin entity known as infinity to serve any self-approving proof imaginable.

it's just nihilism, ya know.

>>11525721
>Proved it wrong already
>"you're trying to sneakily pass the properties of infinity onto a real number."
I'm not sure this counts as a proof anon

>>11525642
m-muh feelings

>>11525733
The state of this board is concerning. If it's not 0.999... = 1 it's this kind of complotist shill.

>>11525733
See >>11525718

>>11525735
[math]S: [0.0, 0.9, 0.99, 0.999, ... ] [/math]
[math]N: [0,1,2,3,4,5,\dots][/math]
N bijects S. Indexing from 0, the n'th term of N is tied to the n'th term of S through the complete sum of [math]\sum_{k=1}^{n} \frac{9}{10^k}[/math], which is also implicitly equivalent to the n'th partial sum in [math]\sum_{n=1}^{\infty} \frac{9}{10^n}[/math].
The 1st element of N is 1. N[1] as n=1 in the sum provides 0.9, which is also the 1st element of S; again, indexing from 0.
All subsequent elements of N and S are tied together, bijected, through the same expectation.
n in N is n in N, and n in S is n in [math]\sum_{n=1}^{\infty} \frac{9}{10^n}[/math]
alternatively, lacking the sum which invokes infinity, N and S are tied by n in N defining the amount of 9's in the n'th element of S.
n=3 (N[3], S[3]=0.999)

does N have an end?
no.
does S have an end?
no.

what comes after S?
is it 1? Or undefined?

What comes after N?
Is it ∞? Or undefined?
it can't be ∞, because ∞ is (NaN).
so maybe asking what comes after N is sensless, for it's (undefined). There is no point to the question, for any number that could come after would also just as easily exist in N.

what comes after S?
is it 1? Or undefined?
It can't be 1 because 1 is the S stand-in for ∞ bijected with N. What's defined for N is also defined for S, then.
so maybe asking what comes after S is sensless, for it's (undefined). There is no point to the question, for any number that could come after would also just as easily exist in S.

every n of N describes an amount of occupied decimal places in every n of S.
If N increases, then the decimal place in S increases. If N increases with an undefined end, then the decimal places of S increase with an undefined end.

Would you say $0.90 is basically a dollar?
No.
Would you say $0.99 is basically a dollar?
Probably.
900 close to 1000?
Not closer than 990.
Not closer than 999.
but these are finite articles. 2 decimals. 3 digits.
Not infinite.

I have an essay for "writing arguments" about a topic mathematicians disagree on due in 3.5 hours. I wanted to switch to a different topic but the prof knew my major and insisted on doing something related to it. He suggested something related to stats or applied math, but I don't think applied mathematicians ever disagree with eachother either. My topic is the different kinds of electoral systems.

The assignment is 3 pages about what they agree on and 2 pages on what they do not agree on. At least I can write the first 3/5 of the assignment. Only two pages will be not make any sense.

>>11525806
>what comes after S?
for example the number 7. also the number 23. maybe even 54. now what does that tell us ?

[math]\frac{1}{10^n}[/math] is a real non-zero number for all real numbers n.
Relative to this fact, how can you say 0.999...=1?

Are you implying there is an unreal amount of 9's?
That you can try counting the 9's, but never finish?
Well you can try counting the real natural numbers and never finish that either, but that doesn't prevent [math]\frac{1}{10^n}[/math] from existing for every real number.

No. What you're implying is that there are definitely more 9's than real numbers. That's why it's an unreal amount of 9's; cause you can't count it with all the real numbers.
Even though you can't even count all the real numbers.

Haha wait so which one is infinity?
The endless nature of counting reals?
Or the endless nature of counting beyond what you already can't count?

gonna need some diddlerkid cuts and crotchy secretions to figure this one out, boys.

>>11525841
"..." means infinite
n is finite

stop pretending finite is infinite

>>11525806
> What comes after S?
> What comes after N?
whatthefuckamireading.jpg

>>11525853
Which one is infinity?
The endless nature of the reals?
Or counting beyond the endless nature of the reals?

If it's the former, [math]\frac{1}{10^n}[/math] exists as a small non-zero number.

If it's the latter, where do you get off assuming you can continue counting AFTER endlessly counting the reals?
that's not allowed you fricken dickhead!

>>11525806
>what comes after S?
0.999....
>What comes after N?
nothing

>>11525864
the entertainment never ends

>>11525864
inf is infinity
sorry to confuse your little brain

>>11525806
>what comes after S?
What do you mean? S is just a set.

There is not even any conclusion in your post, it's not even an argument.

As far as I can tell you are trying to say that 0.999... is not 1 because the set [0.9, 0.99,...] doesn't contain 0.999... but that's an irrelevancy.

>>11525878
Infinity is a retard number for retards who can't comprehend adding 1 to the largest number they can think of.

>>11525884
>inf is number
nope

>>11525841
>Relative to this fact, how can you say 0.999...=1?
Simple: one is irrelevant to the other.

>Are you implying there is an unreal amount of 9's?
Yes, infinity is not a real number.

>That you can try counting the 9's, but never finish?
Yes.

>Well you can try counting the real natural numbers and never finish that either, but that doesn't prevent 110n from existing for every real number.
So? Every nth decimal of 0.999... exists.

>No. What you're implying is that there are definitely more 9's than real numbers.
No, the amount of 9s is the sane as the amount of natural numbers, which is less than the amount of real numbers. The former is aleph null while the latter is 2^(aleph null).

>That's why it's an unreal amount of 9's; cause you can't count it with all the real numbers.
Wrong.

>>11525884
Seethe

>>11525093
Zero is an element of the naturals

>>11525882
Well yes, kinda.
N doesn't contain ∞ because ∞ is not a number. But it contains all other numbers.
N is bijected with S.
S doesn't contain 1, obv. If we assume 0.999...=1, then yes S doesn't contain 0.999..., but it contains all other 0.9 sequences.
How would you write a googol 9's btw?
0.999... works for a googol amount of 9's, right?
I mean i guess it's a finite number but its so big it may as well be infinite.
Oh, is that how this works?
a googol = infinity, so 0.999...=1?

if it was that easy then why are we using these big stupid unwieldly numbers!?

oh wait if googol is infinity that means a googol decimal 9's is definitely 1 so S actually does contain 0.999...

holy fricken crappoli broskerinos. infinity sure is silly.

>>11525721
check this out man
[math] 0.999\dots = \sup\{ 1-\frac{1}{10^n} \mid n \in \mathbb{N} \} = \sup \bigcup_{n\in\mathbb{N}} \{ x\in \mathbb{R} \mid x < 1 - \frac{1}{10^n}\} = \sup \{ x \in \mathbb{R} \mid x < 1 \} = 1[/math] PLS STOP

>>11525917
sup yourself, bruh

pls stop

>>11525916
>If we assume 0.999...=1, then yes S doesn't contain 0.999...,
so what, you're not assuming 0.999... = 1 and that means 0.999... is contained in S?

>>11525916
>How would you write a googol 9's btw?
a googol 9's.

>0.999... works for a googol amount of 9's, right?
What do you mean by works?

>Oh, is that how this works?
No, it's not an approximation.

>>11525357
>Grothendieck famously held a highly controversial view of where mathematics should be headed

Can you elaborate?

>>11525927
Well. Idk if u kno this but ∞ isn't a number.
It's only defined usage in the axiom of infinity is attributing a size to the set of all numbers.
thing was, N was my set of all numbers, so it's size was ∞. Bijected with S, gives me a 0.9 sequence with ∞ size as well.

i guess you can say the size of S is unrelated to the length of the 0.9 sequences, but then how do you even define a length of an element with ∞ when ∞ is a notion of sets and you can't create sets like [math]N^9: [0,9,9,9,9,\dots][/math] mappable to the n'th decimal of set N whose size is infinite, because N9 contains repeated elements which (i'm pretty sure) aren't allowed?

>>11525956
>Idk if u kno this but ∞ isn't a number.
Wrong. https://en.wikipedia.org/wiki/Cardinal_number

Nothing you've said even attempts to disprove 0.999...=1

>>11525956
O wait silly me
[math]S_*: [0.0, 0.9, 0.09, 0.009, \dots][/math]
[math]\lim_{n\to \infty}\frac{9}{10^n}[/math]
just add all the elements of S* together, duh!
S* is an infinite sized set bijected with N which has infinite size. All elements of S added together equal an infinite length element 0.999...

N-nani!?
S*[1] + S*[2] = S[2]
S*[1] + S*[2] + S*[3] = S[3]

so if all elements, inclusive of whatever comes most recently lastly, of S* summed together are an infinite length decimal number, that meanz all whatever comes most recently lastly in S is the same number!

S contains 0.999...
Learn what infinity means.

>>11525987
Kill ursmellf n*ggerbrain

>>11525988
still neanderthal level of explanation, keep evolving

>>11526000
1 day u wil learn how 2 read

I give u an A for Effort.

>>11525988
>S*[1] + S*[2] = S[2]
What do [1] and [2] refer to?

>S contains 0.999...
Wrong.

>>11525998
See >>11525768

>>11525102
I love for the sciences can disagree on shit like this. 0.9999 ad infinitum is obviously not 1. Precision is important to a biochemist like me, but to an inaccurate mathematician, it's close enough.

>>11526013
indices my boy. n's.
set[n] is the indexed element.
S*[1] = 0.9; S*[2] = 0.09
S[2] = 0.99

>>S contains 0.999...
>Wrong.
L O L

>>11525988
{1,2} contains 1 and 2 but not 1 + 3
S* contains 0.9 and 0.009 but S doesn't contain 0.909
no reason to assume that sum of all elements of S* lies in S

>>11526020
How does S*[1]+...S*[n] = S[n] show that 0.999... is a member of S?

>>11525102
>What are limits?

>>11526036
>hurr durr muh infinite approaches
Limits don't exist, retard. Because you can't finish an infinite process.

>>11525956
>∞ isn't a number.
how is that any relevant

if 0.999... isn't in S, then 0.999... is an upper bound for S. however 1 is the smallest upper bound for S. therefore 1 ≤ 0.999... which clearly gives 0.999... = 1.

if 0.999... IS in S, then 0.999... is, well, an element of S. but by definiton. the only elements of S are strings with only finite number of 9's. therefore 0.999... actually has a finite number of 9's! can you tell me what would that number be ?

>>11526039
but you can prove the sloshing remains contained using epsilon-delta

>>11526097
You don't "prove" anything, you just redefine everything and use metaphysically unjustified notions to autistically confuse the reader into believing something that is obviously false.

>>11526029
S* contains 0.9, 0.09, 0.009, etc.
S contains 0.9, 0.99, 0.999, etc
all elements of S* are bijected through the set of all numbers N (which has ∞ elements) via [math]\lim_{n\to\infty} \frac{9}{10^n}[/math], and all elements of S are bijected through N via [math]\sum_{n=1}^{\infty} \frac{9}{10^n}[/math].

Gonna drop the 0 index here, trust me i'm not changing the math. Index from 1.
[math]N: [1,2,3,4,\dots] \\ S: [0.9, 0.99, 0.999, 0.9999, \dots] \\ S* [0.9, 0.09, 0.009, 0.0009, \dots] [/math]
So borrowing from the axiom of infinity, N has ∞ elements. The size of the set of N is ∞. Bijected with S and S* linearly through the sum and lim functions means S and S* are also ∞ size'd

if [math]\sum{n=1}^{k} S*[n] = S[k][/math] is true, lets test k=4
S*[1]+S*[2]+S*[3]+S*[4] =
>0.9 + 0.09 + 0.009 + 0.0009 = 0.9999
S[4] =
>0.9999
K, seems true.
So S* has ∞ elements inherited from the size of N it's bijected with. If all S* elements are added together, S* = 0.999... where infinite size of the set has been translated to infinite length in the decimal. Since sequential S* elements summed to n from the first index also equals the element of S at it's own n'th index, then
>S*[1]+S*[2]+S*[3]+S*[...]-> = 0.999... (with infinite 9's) =
>S[...]-> = 0.999... (with infinite 9's)
So 1 element in S is the contained sum of all S* elements.

this isn't really surprising is it? The set's sizes were infinite to begin with.
I guess the surprising part is the sum.

Figuring to sum all the elements of N,
All elements of N are finite, so summing them all should be a finite number too, even if the process never ended.
All elements of S* are finite, so they too should be a finite number when summed.
And yet this infinite length 0.999... number has finite value???

so, infinity isn't endlessly counting beyond the reals.
Infinity IS endlessly counting the reals.
And [math]\frac{1}{10^n}[/math] exists as a small non-zero real number for every real number n.

0.999... != 1

>>11525093
Wtf do mathematicians even do? Whats the point of being able to do math when computers can do your job way better and faster then you could ever? Whole thing seems more bullshit then even philosophy.

Whether the Collatz conjecture has a counterexample.

>>11526164
Name one person who believes it does.

>>11525228
0.99... is not 1 though, its definitionally smaller then 1 by one infinitismally smaller unit.

>>11525987
>>11526168

>>11526159
[citation needed]

>>11526167
Ted Kaczynski

>>11526103
>You don't "prove" anything, you just redefine everything
Like what?

>and use metaphysically unjustified notions
Like what?

>>11526173
source?

>>11526168
Wrong, retard. This is literally undergrad calculus stuff. If you are too stubborn to give up your stupid opinion on this matter, give me a number strictly between point nine repeating and one.

>>11526195
the bomb ate it

>>11526172
what do you need a citation for, i thought i covered everything in the post chain.

>>11526163

the bottleneck of computers is you have to know what program to write

>>11526159
>Since sequential S* elements summed to n from the first index also equals the element of S at it's own n'th index, then
>S*[1]+S*[2]+S*[3]+S*[...]-> = 0.999... (with infinite 9's)
Which n is [...]?

>So 1 element in S is the contained sum of all S* elements.
You haven't shown that S*[...] is an element of S*. All elements of S* are indexed to a natural number and [...] is not a natural number. You lose, again.

>>11526170
>its definitionally smaller then 1 by one infinitismally smaller unit
If by infinitesimally smaller unit you mean 0, then yes. Otherwise no.

0.999... + 0 = 1

>>11526205
kek
not surprised

>>11526206
How often do they need someone like that? How often does a new math operation come along that need to be programmed into the computer software?
>>11526200
Are you unironically retarded? Thats like saying in terms of whole numbers that 1 and 2 are the same number just because theres no number between them.
>>11526226
No
>0.999... + 0 = 0.999...
Christ I bet you're indebted for life for that shitty math degree and I can still do math better then you lmao

>>11526245
>Thats like saying in terms of whole numbers that 1 and 2 are the same number just because theres no number between them.
dude this shit is older than chuck norris jokes

>>11526245
This looks like bait but it might be that youre an actual retard

>>11526252
>le reddit chuck norris joke because math nigger that spends all his life jacking off to numbers can't comprehend logic
kys

>>11525709

I heard you like infinite processes. Here's one for you.

1 > 0.9
1 > 0.99
1 > 0.999
...
1 > 0.9999...

>>11526258
>n-n-no you're the retard
>has no counter argument
get fucked nigger
0.999... =/= 1

>>11526263
0.99>0.9
0.999>0.99
0.9999>0.999
...
0.999...>0.999...

>>11526263
[math]a_n > b_n \implies \sup\{ a_n\} \geq \sup\{ b_n \}[/math]. Strict inequality cannot be assumed.
[eqn]1 >0.9 \\
1 > 0.99 \\
1 > 0.999\\
\dots \\
1 \geq \sup\{ 1-\tfrac{1}{10^n} \mid n \in\mathbb{N} \} = 1[/eqn]

>>11525093
>What do mathematicians disagree on?
What to do with weebs on The Day of The Rope.

>>11526217
Don't be so hasty.
All elements of S* are natural numbered indices. All elements in total definition of the size of the set are ∞ in size. This is where it's important to make a differention of what infinity means. There are. Asking what the last index of S* is, is meaningless, but we know all elements within S* are finite values (via N), and we know all elements of the set together are ∞ size (also via N)[via axiom of infinity].
There is a weird logic here because "is there a last element"? Not necessarily. But all elements are finite. There's a MOST RECENT element for cases where the sum func of S or the lim func of S* are taken as sequential growth, and that most recent fleetingly-last element will always be a real numbered element with finite value.

Adding all S* elements together nonetheless produces a 0.999... number with infinite length, and the index of the element for whatever final addition occurred, is the same index of S which contains infinite 9's(S*'s sum)

This seems to raise questions elsewhere, like what is the last element of an infinite set or if it's even sensible to assume you can add "all" values in an "infinite" sequence, and an extended notion of that is, if you can't actually have an infinite sum, you can only have a finite (or arbitrarily finite) sum. For any finite sum 9/10^n there exists 1/10^n as a smallest number.

>>11526266
YOU are the nigger with no proof. Show me a number strictly between one and point nine repeating, or shut the fuck up.

>>11526269

lmfao dude, you wrote the sequences to appear to have the same number of terms even though one of them always has an extra 9/10^(n+1)

9/10^n > 9/10^(n+1)

for all n >0

even if we pretended you could give a "size" to a non-terminating sequence, one of those sequences would still always have more terms than the other

|{a1}| < |{a1, a2}|

|{a1, ...}| < |{a1, a2, ...}|

I know you're going to respond to this saying "nuh uh they're equal because MUH BIJECTION" but actually think about this for a moment. if bijections telling us that one set has more elements than another for a finite set, why wouldn't it hold true if there were infinitely many elements?

for example, clearly the following is true

|N U {e}| > |N|


>>11526282

lmfao what? why can't we assume strict inequality from something that is plainly unequal?

You are using a disjunction so you can smuggle in the equality without drawing a contradiction.

>>11526269

|N| < |N U {e}|

>>11526282

You are using a disjunction so you can smuggle in the equality without drawing a contradiction. Very dishonest.

>>11526292
Nah the real bender ends succinctly enough with S* containing infinite elements but all of them are finite [math]\frac{9}{10^n}[/math] real numbers.

That alone means [math]\frac{1}{\infty} \neq 0 & 0.999\dots \neq 1 [/math]

>>11526331
>You are using a disjunction so you can smuggle in the equality without drawing a contradiction. Very dishonest.
Prove [math]a_n < b_n \implies \sup\{a_n\} < \sup\{b_n\}[/math] then

>>11526245
>No
>>0.999... + 0 = 0.999... = 1
Thanks for agreeing with me.

I'm still waiting for that definition that showes 0.999... is infinitesimally smaller than 1 by the way.

>>11526263
4 > 1
4 > 2
...
4 > 4

Brilliant argument.

test.

>>11526365

posets literally do the same thing i just said was the problem in your first argument. please try again.

>>11526379
if it's true, then prove it
if you can't prove it, you can't use it in your arguments

>>11526292
>Adding all S* elements together nonetheless produces a 0.999... number with infinite length, and the index of the element for whatever final addition occurred
There is no final addition since there is no last element as you already admitted. Try again. Thanks for disproving your own argument.

>>11526292
>>11526333
funny that the axiom of infinity basically says infinite sequence arithmetic is (undefined), [math]\frac{1}{\infty}[/math] is (undefined), and [math]0.999\dots \neq 1[/math]

really, nice proof.

what could have been a good thread turned out to be an awful fucking thread

>>11526385
If there is no adding a "final" element, that means you cannot add "all" elements, making infinite arithmetic undefined, which I wouldn't mind cause some fat retarded pajeet did fraud math with infinite arithmetic, and it would mean "infinite sums" are instead arbitrarily finite sums, which also helps support 0.999... != 1.

if you instead first assume that you actually CAN add "all" elements, regardless if there's a knowable final element or not, with the poorly formatted biject function listed here >>11526159
>if [math]\sum_{n=1}^{k} S*[n] = S[k][/math] is true
which i proved it was, then any n'th element of S can be constructed by adding 1->n'th elements of S*

So if "all" elements of S* CAN be added together, this says that there is an element IN S that is equal to the same value, which is [math]0.\overline{999}[/math]

>>11526377

the operands on the right side of my inequalities are being incremented by a fraction with a dividend less than it's divisor and a divisor raised to a strictly increasing natural exponent. your operands are being incremented by a constant amount. false equivalence.

>>11526384

supremum are defined using partially ordered sets which use disjunctions so they can slip in the equality so they can make the trivial argument you are making which is:

a <= a+e

which is trivially true. you can't make your arguments without disjunctions.

>>11526417
true, I want to die

>>11526422
>[math]\sum_{n=1}^{k} S^{*}[n] = S[k] [/math]
hopefully that looks a little more like S* than S multiplied.

>>11525228
Atually not, my boy. While constructive mathematicians agree that given the axioms in standard maths, 0.99...=1, they very much disagree with those axioms.

>>11526395
[citation needed]

>>11526435
Isn't 0.999..=1 also a theorem in constructive mathematics? Even if you don't admit real numbers, some decimal expansions (infinite sums) have limit in the rationals.
I mean I can constructively give you how far you need to go in the sequence to get to a given epsilon to 1.

>>11526422
>>11526434
the fact this is bijected with N, the set of all numbers, which lacks an "∞'th" integer, while the set itself is described as having ∞ size via the axiom of infinity, strictly means that ∞ is not a quantity "greater" than all numbers, but instead is a quantity that is defined by all other numbers.

In essence, the way infinity seemed to be used before was like (number < x < ∞)
while it's proper usage is instead more like (number < ∞).
Although i'm still carrying the greater than symbol here, I'm using it to point to a flaw where it seemed like infinity was separated by two degrees from the reals, and that 2nd degree of implied separation was incredibly ambiguous and a source of retardation.

and in this very same sense, there's no point in asking
(0.999... < x < 1)
you just remove that arbitrary 2nd degree
(0.999... < 1)

>>11526466
if 0.999... is in S and S is in bijection with the natural numbers. then which natural number corresponds to 0.999... ?

>>11526384

remember: the logical equivalent to A v B with only conjunctions and negations is this beauty:

~(~A & ~B)

That's what you're saying every time you use a disjunction.

>>11526422
>If there is no adding a "final" element, that means you cannot add "all" elements
How?

>which i proved it was
You didn't actually prove anything. You stated it's true for any natural number k, which happens to be correct.

>So if "all" elements of S* CAN be added together, this says that there is an element IN S that is equal to the same value, which is 0.999¯
No, it says that IF there is a last element indexed by k in S then all the elements of S* added together sum to S[k]. But we already know there is no last element, so your entire argument is moot.

0.999... is not in S, get over it.

>>11526431
>the operands on the right side of my inequalities are being incremented by a fraction with a dividend less than it's divisor and a divisor raised to a strictly increasing natural exponent. your operands are being incremented by a constant amount.
Distinction without a difference.

0.999... > 0.9
0.999... > 0.99
...
0.999... > 0.999...

Look, the operands on the right side of my inequalities are being incremented by a fraction with a dividend less than it's divisor and a divisor raised to a strictly increasing natural exponent!

>>11525093
Whether pic related is a good book to learn financial mathematics from.

>>11525102

>>11526601
/thread

>>11525464
>>11525452
>>11525253

>>11526515
That's looking at the definition of infinity wrong.
(undefined) is basically the answer to what you're asking though.

infinity is not a quantity greater than all numbers.
infinity is a quantity defined by all numbers, as per the axiom and the proof provided. As all natural numbers are finite and real, there is no point at even which infinite decimals occur that it is still not a finite and real number.

if we take the set of all numbers [1, ->] and map it to the decimal range [1,2] (also applicable to [0,1], but gonna use [1,2] for clarity), there is no greatest number. You can always come up with bigger numbers in the very least by adding +1 to the last number you came up with.
Because there is no ∞'th integer, there is also no application of the existence of 2.0 in the range [1,2], making 1.999... the largest equatable value in that range (and again similarly in [0,1], making 0.999... the largest equatable value in that range).

so 0.999... != 1.

>>11526628

>>11526431
>the operands on the right side of my inequalities are being incremented by a fraction with a dividend less than it's divisor and a divisor raised to a strictly increasing natural exponent. your operands are being incremented by a constant amount.

4 > 0.5
4 > 1.0
4 > 1.5
...
4 > 4.0

Increment = 0.5/1^n

Looks like your distinction is meaningless

>>11526628
>tldr; there's no natural number corresponding to 0.999...
then either 0.999... is not in S or S is not in a bijection with natural numbers
pick one faggot

>>11526551
if "all" elements of S* are added together, it provides cases for all n indexes of S*, which provides cases for all k indexes of S.

[math]\sum_{n=1}^{k} S^{*}[n] = S[k] [/math]

if [math]k \in N [/math], then [math]\sum_{n=1}^{(\in N)} S^{*}[n] = S[(\in N)] [/math]

meaning [math]\sum_{n=1}^{(\in N)} S^{*}[n][/math] is a value inside S [math]= S[(\in N)] [/math]

and if [math]\sum_{n=1}^{(\in N)} S^{*}[n] = 0.999\dots[/math], then 0.999... exists in S.

>>11526567

I am saying this:

|S| < |S U {e}|

The other poster >>11526365 was saying

|S| <= |S U {e}|

which is true, but only trivially so,

YOU are saying

|S| = |S U {e}|

which is blatantly not true under obvious assumptions like {e} not being an element of S prior to the union, etc.

>>11526649
you're asking "whats the biggest number" over and over and over again with circular logic.
the whole point is that there isn't a biggest number.
infinity is not an "element" greater than all the elements of the set of natural numbers.
infinity is the element that describes ALL of the natural numbers.
if you can't think of where the natural numbers end, then you can't think of where infinity ends; but any vertical slice of enumeration within the natural numbers produces a real finite number, and this is the same boundary of infinity, which is why asking "whats the biggest number" is answered with (undefined).
it's literally simple as.
there is no defined biggest number. it's undefined.

>>11526680
>|S| = |S U {e}|
a set and its proper subset can have the same cardinality
I bet you didn't know that lmao HOW FUCKING WEIRD IS THAT

also nobody is talking about cardinality of sets here, only you are. it's not relevant.

>>11526714
you're saying that a set A is in a bijection with a set B
you're saying that an element x is in A
but you're saying that there is no element y in B corresponding to x

it doesn't matter what A, B and x are. there's nothing you can say to make this stop being wrong.

>>11526720

and you are predictably using a bijection to try and claim to differently sized sets have the same size HOW WEIRD IS THAT

>>11526720

Forget set theory for a second and just try to use your fingers. This shouldn't be too hard.

How many items are in this list?

L1 = {a1, a2, a3}

and then how many are in this list?

L2 = {a1, a2, a3, a4}

Which list is bigger? L2, obviously. It doesn't matter how many items you keep adding to the lists, as long as you're adding the same amount to each one, L2 will ALWAYS be bigger than L1. This is so unbelievably simple that it astounds me you genuinely can't understand this. Literally a child could understand this.

>>11525097
>Pi
>has two legs
>Tau
>is twice as much as Pi
>has half as many legs
Why are mathematicians this retarded?

>>11526673
Infinity is used to describe the size of the set of N, not any element inside N, but rather the cumulative count of all individual elements of N.
(k = in N) only describes individual element cases that are a subset of N, not the infinite case which describes ALL of N.
Remember, you require the sum of ALL elements of S* to get an infinite decimal of 0.999..., not just a test of a single element in K.

Saying the sum of ALL elements of S* = 0.999... isn't wrong though, nor is saying it must also exist in S given [eqn]\sum_{n=1}^{k} S^*[n] = S[k] [/eqn]

I think you tried too hard with the proof and it's actually just fine enough to say k=infinity, because infinity is already determined to cover ALL elements inside the set, but ∞ is also not an index location in S in bijection with N so S[∞] means something beyond the scope of the set, unless we assume ∞ as the limit of K is merely all inclusive of all numbers in N, which also says S[∞] is the limit of K and is merely all inclusive of all numbers in S.

It doesn't say which element of S is 0.999..., but it does say 0.999... is in S.

>>11526628
>infinity is not a quantity greater than all numbers.
It is a quantity greater than all real numbers.

>infinity is a quantity defined by all numbers, as per the axiom and the proof provided.
What axiom and what proof?

>Because there is no ∞'th integer, there is also no application of the existence of 2.0 in the range [1,2]
This is a load of nonsense. You can have a mapping from the natural numbers to [1,2] any way you like. For example:

1->1
2->2
3->0.5
4->0.25
5->0.75
Etc.

You will never map to all numbers in [1,2] because they are uncountable. But there is nothing stopping you from making to 2.

>making 1.999... the largest equatable value in that range (and again similarly in [0,1], making 0.999... the largest equatable value in that range).
1.999... = 2 so you're immediately contradicting yourself. Wait, don't tell me you think you can increment through the reals in order... which number comes after 1?

>>11526746
>It doesn't matter how many items you keep adding to the lists, as long as you're adding the same amount to each one, L2 will ALWAYS be bigger than L1
not true
if you add all natural numbers to both sets, they will have the same size

>>11526789
you're old and can't learn new tricks.

everything about your post was dumb and evidence of not reading the chain. I'm not repeating everything that's already been said here just for you cause you're a special little snowflake who doesn't want to read more posts.

>>11526798
answer how is >>11526727 possible lol

>>11525351
0.9* is defined as the limit of a series, it itself has no limit as it is a specific value

>>11526727
huh?
A is in bijection with B
the n'th element of A, if it's x, is equated to the n'th element in B via a sum or lim function.
did you mean to say "element y"?
i don't understand what x and y here are referring too.

>>11526680
>|S| < |S U {e}|
What the fuck is {e}? The empty set? What does this have to do with what you're replying to?

>>11525093
Literally nothing.
Math is rigorous unlike any other 'science'

>>11525093
whether 0 is a natural number or not

>>11526728
>and you are predictably using a bijection to try and claim to differently sized sets have the same size
How do you know they're differently sized if there is a bijection between them?

>>11526746
>How many items are in this list?
>L1 = {a1, a2, a3}
It depends on what a1, a2, and a3 are.

>It doesn't matter how many items you keep adding to the lists, as long as you're adding the same amount to each one, L2 will ALWAYS be bigger than L1.
Incorrect. Adding the natural numbers to both will make their cardinalities equal. You lose again. Learn basic set theory.

>>11526794

Look, I know you are talking about bijections between infinite sets- I am not talking about infinite sets- I am talking about the number of items in a list if you indefinitely add items to them where one list has more items than the next.

This is an indefinite, unfinished process, not a finite object. I'm not asking you to compare "infinite sets", I'm asking you to indefinitely compare the updated size of two lists. One will always be bigger than the other. You can "try this at home." It's not complicated.

The list with more items will always be larger. You can repeat this process indefinitely and it will always be true.

The fundamental difference here is that I am beginning with a *finite* amount, and repeating it indefinitely whereas you are beginning with an infinite "amount", and comparing the two with bijections. You can do that if you want to, but fundamentally you have to understand these are two different things we're talking about.

>>11526828

read
>>11526860

>>11526831

We're talking about different things here buddy. You're talking about a "completed" infinity, I am not.

>>11526827
>finite ordinal/cardinal
>additive identity
Yep, it definitely is.

what 1*1 equals

>>11526828

I am incrementing finite sets, updating and then comparing the size of each set, and simply stating that the comparison does not change regardless of how many iterations of this process you do- you are comparing infinite sets.

we're talking about different things. it's okay. I don't have a problem with set theory. it's an interesting field of study. so it topology and model theory and the likes. But you have to accept that we're talking about different things.

>>11526860
agreed that you've described two different things

HOWEVER

>This is an indefinite, unfinished process
this is what 0.999... is not
>a finite object
this is what 0.999... is

I'll let you figure out the conclusion on your own

>>11526863
>talking about a "completed" infinity
Even /sci/ was never this retarded. What happened?

>>11526779
>but it does say 0.999... is in S.
It doesn't. S*[inf] and S[inf] are undefined so his "proof" fails.

>>11526798
See >>11526014

>>11526881

Okay I'm glad we've found some common ground because this more or less always turns into a needlessly heated debate where opponents talk past each other instead of trying to understand where they differ from each other and what caused it and how it can be reconciled. Nice talking with you.

>>11526863
>read
>>>11526860 #
I did, sets are not a process in time so your entire argument is irrelevant.

>We're talking about different things here buddy. You're talking about a "completed" infinity, I am not.
You're talking about something completely different, I'm on topic.

>>11526880
>I am incrementing finite sets, updating and then comparing the size of each set, and simply stating that the comparison does not change regardless of how many iterations of this process you do-
OK, why?

>>11525093
1^0
0!
0^0
...

>>11526866
yet it's still controversial, and AFAIR, it's not included as natural in some contexts.

>>11526871
1. What other value could it have?

>>11526995
2

>>11526999
Hi Terence. Your show sucks.

>>11526927
>sets are not a process in time

Well maybe it is true that there are sets which are not a process in time, but poor old me, I just happen to a process in time, so I'm stuck thinking about sets like that too. Oh well. Euler did fine without your perspective on sets. Or maybe you aren't satisfied with the mathematics of Euler?

>>11526939

Because that is how I naturally think about indefinite processes of course. I never begin with infinity, I don't know how to do that, or even know what that means. I begin with a finite amount and then increment indefinitely.

If it's good enough for Euler, it's good enough for me.

>>11525652
it seems you misunderstood something. if he claims that infinite sets don't exist, he is not saying that the set of natural numbers is finite but he is saying that there is no set containing all natural numbers

Tried to condense this as best i could but words.
>pt1

The Axiom of Infinity defines infinity as the size of the set of all Natural numbers
[math]N: [1,2,3,4,5,\dots][/math]
Thus N is the set of Naturals that has ∞ individual elements in it, and every element is a finite natural number

[math]S: [0.9, 0.99, 0.999, \dots][/math]
Let S be a bijecting set of N that also solves the partial sums of [math]\sum_{k=1}^∞\frac{9}{10^k}[/math] where k=N[n] (the n'th element of N)

[math]S^*: [0.9, 0.09, 0.009, \dots][/math]
Let S* be a bijecting set of N that also solves [math]lim_{k} \frac{9}{10^k}[/math] where k=N[n]

N, S, and S* each have ∞ elements and thus ∞ size to their sets, with full bijection between each set, indexing from 1.

[math]\sum_{m=1}^{p} S^*[m] = S[p][/math]
Shows that any cumulative m iterative summation of S* elements starting from the first element of S* is equal to the p'th element of S.
[math]p=3;(S^*[1]+S^*[2]+S^*[3])=(0.9+0.09+0.009)=S[3]=0.999[/math]

As S* has ∞ size, and all it's elements are bijective real numbers with N, and all it's elements occupy a unique decimal location, then [math]\sum_{m=1}^∞S^*[m]=0.999\dots[/math], a number with ∞ decimal places which also exists as an element in S, for ∞ is defined as the quality which covers ALL elements of the infinite set N. As ALL elements of N are bijective to S, this further covers ALL elements of S.
[math]\sum_{m=1}^∞S^*[m]=0.999\dots=S[∞][/math]
The index location of 0.999... within S is undefined, as is the largest real number in N also undefined; but 0.999... does indeed exist in S, for ALL of N is infinite.
However, since each element of S is singularly unique and unrepeated, there exists 1 unique case of an element in S equal to 0.999...

>continued

>>11527191
>pt2

The assumption that 0.999... < x < 1 arbitrarily assumes N < x < ∞, which is a false sense of an extra degree of assumed required separation.
Removing the arbitrarily redundant extra degree, we arrive at 0.999... < 1 and N < ∞, for N already contains the notion of an (n+1) larger element than n (N is the set of ALL natural numbers), so S too already contains the notion of an (n+1) larger decimal expansion than n.
As 0.999... is self contained within ALL of N, shown here via bijection, shows a capacity for a true ∞ of decimals between [0,1], and this ∞ is inclusive of 0 (a beginning) while exclusive of 1 (an end), for the largest possible element between [0,1] is 0.999..., an infinite decimal encapsulated within the quantity of elements in N which are all individually finite.

Takeaways:
1. Infinite arithmetic may be undefined, for example: [math](1+2+3+4+...)= undefined [/math]
2. [math]\frac{n}{∞} = undefined [/math]
3. [math]n*∞ = undefined [/math]
4. [math] n \pm ∞ = undefined [/math]
5. 0.999... < 1.0

end

>>11527194
>pt 1

The Axiom of Infinity defines infinity as the size of the set of all Natural numbers
[math]N: [1,2,3,4,5,\dots][/math]
Thus N is the set of Naturals that has ∞ individual elements in it, and every element is a finite natural number

[math]S: [0.9, 0.99, 0.999, \dots][/math]
Let S be a bijecting set of N that also solves the partial sums of [math]\sum_{k=1}^{\infty}\frac{9}{10^k} [/math] where k=N[n] (the n'th element of N)

[math]S^*: [0.9, 0.09, 0.009, \dots][/math]
Let S* be a bijecting set of N that also solves [math]lim_{k} \frac{9}{10^k}[/math] where k=N[n]

N, S, and S* each have ∞ elements and thus ∞ size to their sets, with full bijection between each set, indexing from 1.

[math]\sum_{m=1}^{p} S^*[m] = S[p][/math]
Shows that any cumulative m iterative summation of S* elements starting from the first element of S* is equal to the p'th element of S.
[math]p=3;(S^*[1]+S^*[2]+S^*[3])=(0.9+0.09+0.009)=S[3]=0.999[/math]

As S* has ∞ size, and all it's elements are bijective real numbers with N, and all it's elements occupy a unique decimal location, then [math]\sum_{m=1}^∞S^*[m]=0.999\dots[/math], a number with ∞ decimal places which also exists as an element in S, for ∞ is defined as the quality which covers ALL elements of the infinite set N. As ALL elements of N are bijective to S, this further covers ALL elements of S.
[math]\sum_{m=1}^∞S^*[m]=0.999\dots=S[∞][/math]
The index location of 0.999... within S is undefined, as is the largest real number in N also undefined; but 0.999... does indeed exist in S, for ALL of N is infinite.
However, since each element of S is singularly unique and unrepeated, there exists 1 unique case of an element in S equal to 0.999...

>>11527202
>p2

The assumption that 0.999... < x < 1 arbitrarily assumes N < x < ∞, which is a false sense of an extra degree of assumed required separation.
Removing the arbitrarily redundant extra degree, we arrive at 0.999... < 1 and N < ∞, for N already contains the notion of an (n+1) larger element than n (N is the set of ALL natural numbers), so S too already contains the notion of an (n+1) larger decimal expansion than n.
As 0.999... is self contained within ALL of N, shown here via bijection, shows a capacity for a true ∞ of decimals between [0,1], and this ∞ is inclusive of 0 (a beginning) while exclusive of 1 (an end), for the largest possible element between [0,1] is 0.999..., an infinite decimal encapsulated within the quantity of elements in N which are all individually finite.

Takeaways:
1. Infinite arithmetic may be undefined, for example: [math](1+2+3+4+...)= undefined [/math]
2. [math]\frac{n}{∞} = undefined [/math]
3. [math]n*∞ = undefined [/math]
4. [math] n \pm ∞ = undefined [/math]
5. 0.999... < 1.0

>>11527202
>>11527211
fixed broken carrots.

>>11526004
I'll put your A as a gift in a museum of evolution. I'll label it as the gift from a prehistoric ape trying to communicate with 21st century humans. Thanks, ook iik aak.

>>11527433
See >>11527266

>>11527016
>Well maybe it is true that there are sets which are not a process in time, but poor old me, I just happen to a process in time, so I'm stuck thinking about sets like that too.
So you must think sets are humans, right? Hint: what you think a set is is not the basis for a mathematical argument. What it is defined to be is.

>Euler did fine without your perspective on sets.
Who cares?

>>11527164
How many natural numbers can a set have?

>>11526601
>>11525102
>>11525228
all wrong. this is an artefact from our base 10 number system.

If an intergalactic alien species with superior maths came to earth they would disagree

>>11527202
>Shows that any cumulative m iterative summation of S* elements starting from the first element of S* is equal to the p'th element of S.
It doesn't, since any such summation of S* elements includes the summation of all elements, 0.999... which is not a p'th element of S. All you're doing is assuming the false statement you were supposed to be proving.

>a number with ∞ decimal places which also exists as an element in S, for ∞ is defined as the quality which covers ALL elements of the infinite set N.
This is gibberish and doesn't follow. Inf is not defined that way and is not an element of N by definition.

>>11527211
>The assumption that 0.999... < x < 1 arbitrarily assumes N < x < ∞, which is a false sense of an extra degree of assumed required separation.
Incorrect gibberish. Is N supposed to be a number now when before it was a set? What implies that x is greater than this new made up number? Just stop posting, retard.

>>11527689
https://www.wolframalpha.com/input/?i=convert+0.9...+to+base+2

>>11527211
1+2+3+4+... = inf
n/inf = 0
n*inf = inf
n+inf = inf
n-inf = -inf
0.999... = 1

>>11528051
>retardnigga cant read
y i k e s

>>11528069
https://www.wolframalpha.com/input/?i=1%2B2%2B3%2B4%2B......
https://www.wolframalpha.com/input/?i=10*inf
https://www.wolframalpha.com/input/?i=10%2Binf
https://www.wolframalpha.com/input/?i=10-inf
https://www.wolframalpha.com/input/?i=0.999......

>>11528069
https://www.wolframalpha.com/input/?i=1%2B2%2B3%2B4%2B......
https://www.wolframalpha.com/input/?i=10%2Finf
https://www.wolframalpha.com/input/?i=10*inf
https://www.wolframalpha.com/input/?i=10%2Binf
https://www.wolframalpha.com/input/?i=10-inf
https://www.wolframalpha.com/input/?i=0.999......
https://www.wolframalpha.com/input/?i=10%2Finf

>>11528069
https://www.wolframalpha.com/input/?i=1%2B2%2B3%2B4%2B......
https://www.wolframalpha.com/input/?i=10%2Finf
https://www.wolframalpha.com/input/?i=10*inf
https://www.wolframalpha.com/input/?i=10%2Binf
https://www.wolframalpha.com/input/?i=10-inf
https://www.wolframalpha.com/input/?i=0.999......

>>11528149
https://www.wolframalpha.com/input/?i=sum%5B9%2F10%5En%2C+%28n%2C1%2C10%5E308%29%5D

anyway, i showed a proof that can be solved. You're just linking some old garbage information

>>11528154
forcing WA to fiddle with floating point numbers is retarded, so what

>>11527700
infinity is indeed not an element of N.
Didn't you read the proof?

Infinity is defined as the size of the set of natural numbers N.
S* is a set bijected with N for every element of N, meaning S* also has the same infinite size.

Adding all of S*'s elements together thereby produces a 0.999... decimal number with infinite decimal 9's.

and this formula [math]\sum_{m=1}^{p} S^*[m] = S[p][/math] shows that ever summation of elements of S* produces a term which exists inside S.

following it now?
it means that 0.999... infinite decimal number also exists inside S, within the bounds of infinity.

>>11525616
>your argument is not formal
None of the objections to .999...=1 are formal either. Such rigor is not necessary and may even be counterproductive in some circumstances

>>11527666
about 1000. after that the numbers just get too big

Lots. To name a few:

1)P=NP
2)Does infinity exist? If there is a smallest number, there must be a largest number

>>11525351
no
real numbers can be expressed with convergent sequences of rational numbers, and the differences between real numbers can be expressed with convergent sequences of rational numbers. When the rational number sequence representing the difference between two real numbers has a limit at 0, then the real numbers belong to the same equivalence class. When discussing the real numbers, this equivalence class is called "equality," or "=". So 0.999... = 1 in the real numbers.

>>11525658
Yes it does, retard. If the set of Natural Numbers, N, isn't infinite, then it is finite. If it is finite, then, combining that with the well-ordered principle, it must have a maximum, call it M. What is it?

>>11526245
No, retard. The reals, unlike the Natural Numbers, are dense. Thus, if point nine repeating and one aren't the same, then it is MATHEMATICALLY IMPOSSIBLE for there to not be an INFINITE number of numbers between them. Just pick 1 of them.

>>11528287
>If there is a smallest number, there must be a largest number
neither exist

>>11528323
>INFINITE number
infinite amount
inf isn't a number, let's not confuse the unwashed even more

>>11528422
>inf isn't a number,
And yet you're saying it's an element of N. Interesting

>>11528445
-look there are 15 ducks in the box
-oh you're calling 15 a duck? interesting
hurrr durrrrrr

>>11525360
>precise
>his dfefinition is wrong
It should be 1-10^{-n}.
Your sequence wouldn't converge.

>>11525676
Imagine falling for there bait... You can't teach someone who doesn't want to be taught.

>>11528546

>>11527202
>ALL elements of N are bijective to S
>0.999... does indeed exist in S
>0.999... = S[∞]
∞ is in N

>>11528576
1 is in N

mindblown.jpg

>>11525683
In what branch of mathematics is law of excluded middle not accepted? Is it even possible to work without it?

>>11528743
Mostly just the crackpot branch.

>>11525093
Axioms and whether or not proofs of certain theorems are valid.

>The virgin axioms vs the Chad affirmations

>>11528165
>infinity is indeed not an element of N.
Then S[inf] is meaningless.

>Infinity is defined as the size of the set of natural numbers N.
That's incorrect. The axiom of infinity doesn't define infinity as anything. The axiom of infinity merely says that a set exists containing all of the natural numbers. This is like saying, "a chihuahua exists, therefore dogs are defined as chihuahuas."

>and this formula ∑pm=1S∗[m]=S[p] shows that ever summation of elements of S* produces a term which exists inside S.
You keep repeating this without proving it. Again you're assuming what you needed to prove.

On the left hand side, p can be any natural number or inf. On the right hand side, p can only be a natural number. So the equation does not hold for all summations, for all values of p. In order to show that it does, you would have to show S[inf] is defined, which is what you claimed to have proved simply by writing out the equation. How many times do I have to explain this circular logic before you respond to it?

those people who have done away with the infinitesimal are retarded. 0.9999999... != 1. It is an infinitesimal value less than 1. 0.333333... != 1/3. It is an approximation, but you never get a conclusion. It is an infinitesimal value less than 1/3.

>>11529561
>infinitesimal
Yawn. Nobody takes that serious, it's been over 100 years since it was discarded and was never rigorous to begin with...

Quite literally bullshit.

>>11529582
This is the same pretentious attitude of people who think that if the limit of f(x) as x tends toward y = something, then f(y) = that thing, even if f(y) is undefined.

It's just not true. That is, in particular, the very vice you complain about. Not rigorous enough.

>>11529561
no daylight between the infinitesimal and zero
1/inf=0

>>11529561
define 0.99999....

>>11529615
>no daylight between the infinitesimal and zero
>1/inf=0
Same problem as I described in >>11529601.
Just because as x gets larger and larger, (1/x) gets closer and closer to 0 doesn't mean that 1/inf = 0. That's why it's called a 'limit', and not just the answer to the equation at that point.

Say you had the function f(x) = 1/x. As x gets closer and closer to 0, then 1/x gets closer and closer to infinity. Would you say that 1/0 = inf, or just that the limit of 1/x as x approaches 0 = infinity? Why or why not?

>>11529619
Can be defined many ways. One way is
0.9999... = 1 - 0.000...1
0.9999... = 9/10 + 9/10^2 + 9/10^3 + 9/10^4...

As I mentioned earlier, it is most commonly found because it's what you get when you multiply the decimal expression of 1/3(But not the reality of the fraction) by 3.

>>11529601
>This is the same pretentious attitude of people who think that if the limit of f(x) as x tends toward y = something, then f(y) = that thing, even if f(y) is undefined.
LOL, this is true, but totally unrelated. Also limits replaced infinitesimals, so chose one, not both.

Infinitesimals are a dead theory. No mathematician has used them for a hundred years.
We have analysis now, no need for shitty rigourless bullshit like infinitesimals.

>Would you say that 1/0 = inf, or just that the limit of 1/x as x approaches 0 = infinity? Why or why not?
It's a common convention when working with the one point compacitification of the reals,but obviously non sensical when you talk about the reals.

>>11529647
>=
Please define this symbol.

>>11529647
>0.9999... = 1 - 0.000...1
define 0.0000....1
>0.9999... = 9/10 + 9/10^2 + 9/10^3 + 9/10^4...
define what do the dots mean. do you mean a limit of something ?

>>11526761
Clearly the solution is to have 4 circle constants: the tau symbol represents pi/2, pi stays the same, pi with 3 legs represents 3pi/2, and pi with 4 legs represents 2pi.

>>11529629
so tell us what's between the infinitesimal and zero

>>11529649
>limits replaced infinitesimals, so chose one, not both
You have a very unorganized way of thinking about these things, don't you?
Do you see that you /could/ use both, and you ask me to chose one and not both because of a notion about the history of mathematics you have, or do you think that using both is impossible and/or contradictory?

>Infinitesimals are a dead theory. No mathematician has used them for a hundred years.
>We have analysis now, no need for shitty rigourless bullshit like infinitesimals.
This seems like more of you nerding out about what you think the history of mathematics was. Do you have any actual, articulable problem with infinitesimals, or is it just that you wanna be in the in-group?

>It's a common convention when working with the one point compacitification of the reals,but obviously non sensical when you talk about the reals.
I'm not sure I understand. Are you saying that 0.999... = 1 is convention, and what we're really saying is that 0.999... might as well be 1, because it approaches 1?

>>11529681
>Do you have any actual, articulable problem with infinitesimals
They are ill defined and in no way formalized or rigorous.

>Are you saying that 0.999... = 1 is convention
No, I said 1/0 = infinity in the one point compactification of R, that is the convention.

0.999...=1 is true by the definition of the real numbers.

>0.999... might as well be 1, because it approaches 1?
That is basically the definition of equality in the real numbers.
Although it's usually phrased as them being in the same equivalence class of cauchy sequences.

>>11529654
That the thing on the left represents the same value as the thing on the right.

That is to say, that if in any context you had separable, continuous values, and you were to represent both sides of the "=" sign, carrying out all of the process described to it's completion(Which is only theoretical in the case of infinite processes described with a "...", but with the fairly simple ones' we're talking about, you can prove pretty easily what must or musn't look like in the end), following a consistent convention for things like the order of operations, you'd get to the same result.

>>11529692
>That the thing on the left represents the same value as the thing on the right.
Wrong. Want to try again? Or should I give you the answer?

>>11529679
They are not classed such that the question you are asking makes sense. Just as 1/inf isn't something you can seriously notate. What would it mean if you could algebraically deal with infinity like it was a number? 1/inf = 0 -> 1 = 0, the same way you're trying to say (The amount in-between infinitesimal an 0) 1/inf - 0 = 0 -> 1/inf = 0.

If you are to work with values like this, you have to put care that the algebra is sensible.

>>11529709
>Just as 1/inf isn't something you can seriously notate.
You can and you do. Open any analysis textbook and somewhere they will make that convention, usually for Lp spaces.

>What would it mean if you could algebraically deal with infinity like it was a number?
That you don't end up with a field, but that doesn't matter in many situations.

>>11529709
see >>11529656

>>11529691
>They are ill defined and in no way formalized or rigorous.
What is your standard for definition? Is it that smart people use a lot of greek symbols?

>No, I said...
Oh, ok. I didn't know what you were saying exactly, so that's why I asked.

>That is basically the definition of equality in the real numbers.
>Although it's usually phrased as them being in the same equivalence class of cauchy sequences.
If that's what "equivalence class of cauchy sequences" means, and that's how equality of real numbers is defined, then I wonder who set about that standard and why. It certainly isn't something you can take granted for functions, for the same reason that the `one point compactification of R` is only convention, and not an absolutely exclusive truth.

>>11529700
>Wrong
Wrong! Hehe hoho, I see why you do this, it's fun!
Do you need help reading?

>>11529647
What is 1/3-0.333...?

>>11529721
>What is your standard for definition?
Your theory of real numbers should be formally derived from a set of reasonable axioms.

>If that's what "equivalence class of cauchy sequences" means, and that's how equality of real numbers is defined, then I wonder who set about that standard and why.
I believe it was Cantor, for the who. For the why, it was because he wanted to rigorously define the real numbers for the first time.

>It certainly isn't something you can take granted
If you work with real numbers you do. Since everybody uses Cantor's, or an equivalent, definition.
If you want to make your own definition, fine, but don't bother other people with it.

>for the same reason that the `one point compactification of R` is only convention, and not an absolutely exclusive truth.
Not really comparable. That is just a way to avoid case distinctions.

>>11529725
Do you want the correct answer? Yes or no?

>>11529656
>define what do the dots mean. do you mean a limit of something ?
I admit, I was being unscrupulous with my use of the dots this time.

In 0.9999... = 9/10 + 9/10^2 + 9/10^3 + 9/10^4..., the first thing is the limit of the second thing.

But the second thing is not the limit of the first thing, whereas the first thing(0.9999..>) represents the decimal expansion of 0.9999, with an arbitrarily large(And, in the theoretical space, infinite) number of digits.

>>11529750
Can you give me the correct answer to this question first, so I know whether to answer yes or no?

Or, well, I'm a bit nervous about upsetting you. Can you give me the correct answer about whether or not I'd upset you if I ask questions?

>>11529760
>the first thing is the limit of the second thing.
>But the second thing is not the limit of the first thing
that's now how equality works, man

>>11529773
I just mean to say that the dots in the second thing represent that we're taking the limit of a serious, and that the dots in the first thing represent something different.

>>11529779
of a series*

>>11529765
The answer is two real numbers are equal if they are the same equivalence class if cauchy sequences.
Since generally you actually calculate with representatives, instead of equivalence classes, two representatives are equal if their difference goes to zero.

>>11529760
>the first thing is the limit of the second thing.
Nice freshman post. The first thing is the limit of the partial sums of the second thing. They're equal.

>>11529779
so you're saying that the second thing is a limit of a series. great.
it's absolutely irrebuttable that the limit of this seris is 1 though.

>>11529788
>>11529786
Oh, you got me. I couldn't have meant that.
The limit is the thing it approaches, and it does get closer and closer. But it never /gets/ there. There must be a way to talk about that.

>>11528743
Constructive mathematics
>>11528761
No, constructive mathematics is not "crackpottery" it's motivation is quite noble desu despite the fact that I prefer a classical view.

>>11529793
Sure, mathematically you can say that the rational number 1 is not the member of the sequence (0.9, 0.99, 0.999,...) which is trivial and also completely irrelevant to the discussion.

>>11529793
>The limit is the thing it approaches, and it does get closer and closer. But it never /gets/ there.
What process in time are you describing?

>>11529797
But whatever you get to during the decimal expansion of 0.999... must be in the sequence (0.9, 0.99, 0.999, ...) because moving up in that sequence is the very meaning of 'decimal expansion'. Just because the sequence is arbitrarily large, and therefore gets arbitrarily close to 1, and therefore approaches it in the limit-sense, doesn't mean you can say it = 1, because that is to say that something about decimal expansion would lead it to be 1, and that isn't true.

>>11529805
It's not temporal in the sense we have time, but logical time. Like how in the syllogism
All men are mortal
Socrates is a man
Socrates is mortal

"All men are mortal" comes before "Socrates is mortal"

>>11529806
>>11529793
0.999... is NOT a sequence
0.999... is a real number. otherwise 0.999... != 1 trivially, because the RHS and LHS wouldn't even be the same type of object.
if 0.999... is a real number, what more reasonable definition can you think of than the limit of the series

yes, the sequence 0.9, 0.99, 0.999, ... approaches 1, and is never equal to it. but 0.999... is not this sequence, it's the number that's being approached.

>>11529806
>Just because the sequence is arbitrarily large
The sequence isn't arbitrarily large, it only has one size. All you're doing is stopping the sequence at an arbitrary point and pretending that that stopping point is the sequence. Just like you pretend a partial sum is the sum itself. That tactic won't work, so give it up before you embarrass yourself further.

>>11529808
The sum is immediately 0.999... in logical time.

>>11529816
>0.999... is NOT a sequence
>0.999... is a real number.
A real number is, by definition, an equivalent class of sequences.

More specifically, 0.999... and 1 are both understood to be *representatives* of equivalence classes, so they are indeed both sequences, they are also in the same equivalence class, so they equal one another.

>>11529816
>0.999... is NOT a sequence
It's a Cauchy sequence.

>yes, the sequence 0.9, 0.99, 0.999, ... approaches 1
No, the members approach 1.

>and is never equal to it.
The sequence as a whole, represented as a real number, is equal to 1.

>>11529822
I'm not wrong.
0.999... is not a sequence.
0.999... is an /equivalence class/ of a sequence (which is the definition of a real number).

I'm not the one who started being a dick.

>>11529829
>0.999... is an /equivalence class/ of a sequence (which is the definition of a real number).
Read further. Usually you calculate with representatives. 0.999... is a representative, thus it is a sequence, just like 1 is a sequence.
And their difference approaches zero, so by definition they are in the same equivalence class, so by definition they are equal.

>>11529835
Not really correct.
0.9999... is an expression that means the real number, that is, the equivalence class of Cauchy sequences, that is represented by the sequence (0.9, 0.99, 0.999, ...)

>>11529388
>>11529426

>>11529817
>The sequence isn't arbitrarily large, it only has one size
Correct. I was once again imprecise with my language.
>. All you're doing is stopping the sequence at an arbitrary point and pretending that that stopping point is the sequence
I was failing to consider the difference between something being continued until the end, which although it never occurs can be talked about in theory, and the thing being any largeness in particular(That is, that we might only be able to talk about things not regarding the preciseness of the approximation as a part of the argument, which still may be true, but for different reasons).
>Just like you pretend a partial sum is the sum itself. That tactic won't work, so give it up before you embarrass yourself further.
The complete sum can't be 1 for the same reason it can't be 10. It never pushes its way such that it would break being represented by only 9's.

>>11529841
Writing down representatives or equivalence classes is, you know, equivalent, that's the point.

>>11529835
I'm not gonna argue with you over this
I'm 99% sure that you've just misunderstood the point of my post and I could try to explain it to you, but let's not waste our fucking time and move on

>>11529854
Okay, I just think you can't read.

Look, let the expansion of 0.999... be a function.
f(1) = 0.9,
f(2) = 0.99,
f(3) = 0.999.
What is f(inf)? A: Undefined. It would have 0.999 and the 9s would just keep going, but you can't input infinity into a function. It's not really a number.
What is the limit of f(x) as x approaches infinity? 1. Because as it gets more and more .9s, it gets arbitrarily close to 1, more and more as x gets higher.

That doesn't mean that f(inf) = limit of f(x) as x gets closer and closer to infinity.

if you follow, that is to say
0.999... repeating is a description of the endpoint of a process that never ends. Such a description is either incomplete, or oversimplified. BUT
1 is what the end-point can be used as, because the conception is that it's 'infinitely close' to 1, but because of the way we deal with values in modern mathematics, 'infinitely close' is the same thing as saying there is no difference at all.

'infinitely close' can be useful for some things. It shouldn't be thrown away, but ignoring it can also be useful for a lot of things.

Are these AI bot?

>>11529874

>>11529709
so nothing huh
just the usual schizo hand waving
1/inf=0

>>11529894
Multiply by infinity on both sides. 1 = 0 ?
OH? YOU CAN'T DO THAT? WHY NOT?

>>11529946
for the same reason you can't

0*1 = 0*2

divide by zero on both sides to get

1 = 2

What's up with this thread's bump limit?

>>11529949
You can't divide by 0 because 0 doesn't comply with the meaning of division in a very basic way.
Does infinity not comply with the meaning of multiplication?

>>11529953
it doesn't when the second factor is zero

>>11529848
>The complete sum can't be 1 for the same reason it can't be 10. It never pushes its way such that it would break being represented by only 9's.
It immediately is 1, I'm not sure what you're even trying to say at this point. You're stuck on points inside of a process instead of evaluating the entire thing.

>>11529958
>>11529958
Why not? It's well established that for any number x, 0*x = 0. Unless infinity can't be used like a number in that context. Which is exactly what I said and you called schizo hand-waving.

>>11529983
it's also well estabilished that x/x = 1 for any number. does that mean 0 is not a number ?

>>11529981
If you have a finite series (9x/10 + 9x/100 + 9x/1000), then you will never get to x. But when it's infinite, there is a difference? Why is that?

>>11529865
>Look, let the expansion of 0.999... be a function.
What does that even mean?

0.999... is the decimal expansion of 0.999...

>What is f(inf)? A: Undefined.
Only in the sense that you arbitrarily defined f without defining f(inf).

>It would have 0.999 and the 9s would just keep going, but you can't input infinity into a function.
You can if you define the function on it, which you just did by saying f(inf) would be 0.999...

>That doesn't mean that f(inf) = limit of f(x) as x gets closer and closer to infinity.
You just said it would be.

>0.999... repeating is a description of the endpoint of a process that never ends.
What process? A function is not a process and you haven't even shown your function is necessary to describe 0.999...

It's funny how the longest posts are written by those with the least to say.

>>11529989
>it's also well established that x/x = 1 for any number
(0 and 1 aren't numbers, but we'll pretend they are because it's easier that way)
Yes, for any number that you can divide by, when it's divided by itself, you get 1. The reason why it's "any number that you can divide by" is because there is at least one which you cannot divide by, for reasons clear if you break down the concept of division. Do you have such basic conceptual reasons that mean you cannot multiply by infinity, but you can divide by it?

>>11529998
>You just said it would be.
no i didn't, I just said that it's undefined.
Something that is undefined isn't equal to the limit as it approaches the undefined value. How do people get so mentally deficient?
>What does that even mean?
>0.999... is the decimal expansion of 0.999...
I mean it in the simplest term I could mean it.
What we have is the symbol "0.999" and an elipses. I am trying to suss out the meaning of the elipses, so I don't assume what it means already, I try to put it in explicit terms. A function seemed appropriate.

>You can if you define the function on it, which you just did by saying f(inf) would be 0.999...
that's called assuming the consequent

>>11529993
This is like asking why two thirds doesn't equal 1 but three thirds does. Because two thirds is too small.

>>11529952
Yeah, suspicious.

>>11530002
operations with ∞ are mere conventions which are used because it makes our life easier.
you cannot introduce the symbol ∞ and define ALL possible operations with it such that ALL laws of arithmetics will still hold.
so people define SOME operations such that MOST laws of arithmetics still hold.

if ∞ is a number is an unsubstantial question, because it's only semantics. "number" is an element of the set of real numbers. ∞ isn't an element of the set of real numbers, therefore ∞ isn't a number. that doesn't mean that you are forbidden to define operations with it.

>>11530005
>no i didn't, I just said that it's undefined.
You said it's undefined because you can't input infinity into a function, which is false:

>What is f(inf)? A: Undefined. It would have 0.999 and the 9s would just keep going, but you can't input infinity into a function.

>Something that is undefined isn't equal to the limit as it approaches the undefined value.
0.999... is defined.

>I am trying to suss out the meaning of the elipses, so I don't assume what it means already, I try to put it in explicit terms.
The meaning of the ellipses is repeating without end. No need for a function, which doesn't clarify anything.

>that's called assuming the consequent
You mean affirming the consequent? No, it's not even that. Retard.

>>11525498
Desu OwO
[math]B \subset A \Leftrightarrow (\forall{X}(x\in{B}\Rightarrow{x\in{A})[/math]
Thewefowe, a set iz a subwset ofw itzelfw UwU

>>11530043
check me out

let [math]f \colon \mathbb{N} \cup \{ \infty \} \to \mathbb{R}[/math] be [math]f(n) = \frac{1}{n}[/math] for [math]n \in \mathbb{N}[/math] and [math]f(\mathbb{\infty}) = 0[/math]

then [math]\lim_{n \to\infty}f(n) = f(\infty)[/math]

>>11530043
>affirming the consequent
No, I don't. I mean assuming the consequent. Or, if you want to put it in words you've heard elsewhere, because you don't understand english, "begging the question"

>>11525362
Why is axiom of choice controversial?

>>11530111
Because it allows for non-constructive proofs, i.e. proofs where an object can be proven to exist without being able to explicitly define or construct that object.

>>11530065
Lewt me fix de Lawtex OwO
[math]A\subset B\Leftrightarrow\left(\forall X\left(X\in B\Rightarrow X\in A\right)\right)[/math]

>>11529946
because inf/inf and 0*inf are undefined

>>11530181
dasu dasu you fucked it up
and also CAREFUL WITH THAT UNIVERSAL QUANTIFIER EUGENE

>>11530135
I've read a bit, and I understand now. So, does AC, in your opinion, prove or disprove free will?

>>11530108
>I mean assuming the consequent.
Assuming the consequent is the same thing as affirming the consequent. See https://www.uwsp.edu/cols/faculty/Dona_Warren/Documents/Symbolic%2520Logic/WhatIsFormalLogic-Narration.pdf

>Or, if you want to put it in words you've heard elsewhere, because you don't understand english, "begging the question"
Funny how you say I don't understand English when you're demonstrably phrasing things incorrectly. And how exactly is this "begging thre question?"

>>11530111
just like the continuum hypothesis, it is disconnected from the real world
you can assume either that they're true or that they're false or that you don't give a shit, and everything will still work out fine in any real-world based calculation

>>11530237
But the non real world is the only interesting part of maths

>>11530229
>assuming the consequent is the same thing as affirming the consequent
(Flashback)
>>that's called assuming the consequent
>"You mean affirming the consequent?"
lol

Stop speaking in jargon and actually use english. What does "assume" mean? What is the "consequent"? Do I have to spell it out for you, or have you figured it out yet?

>>11530204
why? x/0 is undefined because of the nature of division. What about the nature of infinity or division or multiplication means inf/inf and 0*inf would be undefined? And it doesn't apply for the general case of x/inf (Or else the point he >>11529894 was making would be moot in the first place)?

>>11527664
>you must think sets are humans

you're being silly

>Who cares?

Look guy, if you "don't care" about how Euler thought about math, then we really don't share the same values. I read the masters, not the pupils.

>>11530365
read >>11530020 and also https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations

>>11530323

this is a predator/prey relationship. these guys are narcissists gaslighting sensible people for the fun of it. if they play dumb it's insincere.

>>11530323
>lol
What is funny about that?

>Stop speaking in jargon
You're the only one making up vague phrases with nonstandard meaning.

>What does "assume" mean? What is the "consequent"?
You realize that affirming the consequent assumes the consequent, right?

And you failed to explain how this is begging the question.

>>11530378
>you're being silly
The silly consequences of your nonsense does not reflect on me.

>Look guy, if you "don't care" about how Euler thought about math, then we really don't share the same values.
We don't. How is it relevant to the topic at hand?

Arguing based on what you feel to be true and on what you think long dead men would approve of is no way to go through life son.

>>11530618

3/10

>>11530732
0/10

>>11530773
1/3