/sci/ - Science & Math

[Anonymous No.12438345]

brainlet here. Can someone explain to me what a "real number" is?

[Anonymous No.12438358]

>>12438345
It's a number that's real. As opposed to the set of fake numbers, which we don't talk about as they're not real.

[Anonymous No.12438360]

wtf is with this shit lately.

[Anonymous No.12438361]

>>12438358
So is that a rational number? I've heard people say some irrational numbers are also real. What do they mean by that?

[Anonymous No.12438366]

>>12438345
It's a topological extension of the rationals. There is a so-called Completeness axiom that "plugs" all the "holes" in rationals with irrationals to create a continuous real line.

[Anonymous No.12438371]

>>12438345
The equivalence class of Cauchy sequences in the rational numbers with the equivalence relation a_k R b_k iff
lim_{k-> infinity} |a_k - b_k| = 0
Note that most of these sequences do not converge in the rationals

[Anonymous No.12438372]

please using yahoo

[Anonymous No.12438373]

>>12438366
>There is a so-called Completeness axiom that "plugs" all the "holes" in rationals with irrationals to create a continuous real line.
Can you explain to me this bit? How do you "plug the holes"? What does that even mean?
Sorry I'm a brainlet, this is all very confusing to me.

[Anonymous No.12438378]

>>12438371
Hi thanks for the reply. Can you explain to me what you mean by a "Cauchy sequence", specifically the "sequence" part?

[Anonymous No.12438391]

>>12438361
Yes, irrational numbers are real, transcendentals as well, but there's no way for us to conceivably reach them in our reality, only approximate them really well.

[Anonymous No.12438394]

>>12438373
It's a very technical axiom. We're basically saying that there are numbers that are sandwiched between rationals, but aren't rational themselves like [math] \sqrt{2} [/math]. The Completeness axiom asserts that these numbers are part of the set of the reals, that is they "exist". So for instance let's say you have some interval of rational numbers [math] (0, \sqrt{2}) [/math]. This set does not have what we call a supremum in the rationals (a fancy name for a maximum, but it doesn't have to be a part of the set). The completeness axiom states that while the supremum may not exist in the rations, it always exists in the reals. This is what I meant by plugging the holes.

[Anonymous No.12438401]

A sequence is a function with the naturals as domain. A sequence of rational numbers is an infinite ordered list of rational numbers.
A sequence is Cauchy if the "tail" of the sequence gets arbitrarly close. I mean if you give me a small positive epsilon I can tell you a natural number that all the elements after this nuber will be epsilon close of each other

[Anonymous No.12438402]

start a reals general faggots

[Anonymous No.12438405]

>>12438345
First things first. The set of all rational numbers is called [math]\mathbb{Q}[/math]. A rational number is any number that can be expressed as a fraction of two integer numbers (think like -2/3 or 4/2).

Turns out that some numbers cannot be expressed as a fraction. For example the number p such that [math]p^2=2[/math] cannot be expressed as a fraction. These numbers are not rational. If you consider the set that unites these numbers with the rational numbers, you get the set of all real numbers.

[Anonymous No.12438406]

>>12438378
>>12438401
Thats my awnser

[Anonymous No.12438422]

>>12438405
You need to be more specific than that. By your definition a root of [math] p^2 = -2 [/math] is real, because it cannot be express as a fraction. The crucial detail is set ordering. We know that there exist rational numbers p and q such that [math] p < \sqrt{2} < q [/math], which you can't do with [math] p < i\sqrt{2} < q [/math]

[Anonymous No.12438426]

>>12438391
Very interesting. But how do we know they're real?
>>12438394
You mention the number sqrt(2). But do we really know it exists? I've read that no rational number squares to 2, so to get such a number we need to go outside of rational numbers, yeah?
Can we prove this completeness axiom?
>>12438401
>A sequence of rational numbers is an infinite ordered list of rational numbers.
Interesting. Of course we can't write down infinite lists, so we need to provide some kind of description of such a list? What's the canonical form in which all such sequences can be written down? Do you give an algorithm in some programming language?

[Anonymous No.12438429]

>>12438405
>For example the number p such that p2=2p2=2 cannot be expressed as a fraction
How do you know such a number p exists?

[Anonymous No.12438437]

>>12438426
>But do we really know it exists?
If you browse /sci/ you will find plenty of schizo posts about *insert_set_name* doesn't exist, (((they))) are lying to us REEEEEEEEEEEE!!!!
The truth is, whether or not something "really" exists is a question of ontology, and math doesn't deal with it. "Existence" in mathematics is just a statement on that you can find a certain element in a set that satisfies the given statement.

[Anonymous No.12438448]

>>12438437
>"Existence" in mathematics is just a statement on that you can find a certain element in a set that satisfies the given statement.
How is that different?
What if I asserted the existence of a set of all describable numbers. Wouldn't I need to prove its existence, or would that be a question of philosophy.
Also isn't it true that a lot of open questions in mathematics are questions about existence?
Why is it sometimes necessary to prove something exists but not other times? Sorry this is very confusing.

[Anonymous No.12438453]

>>12438429
It's square root of 2. It exists as the solution of an equation, can be proved to be not rational.

[Anonymous No.12438459]

>>12438448
different from what? I didn't give you a definition of what is real from ontology. Existence in mathematics is a statement about the content of your set. It doesn't have anything to do with irl existence. I can have a set A that contains three unicorns, one pink, one white, and the other one black. There definitely exists an element in that set which satisfies the property that its black. Does that mean unicorns exist?

[Anonymous No.12438460]

>>12438453
>It exists as the solution of an equation
How do you know this? Is that because all equations have solutions?

[Anonymous No.12438467]

>>12438459
Hmmm, wait, so do we in mathematics just take the existence of real numbers as an axiom and go from there? Why do people think the reals don't exist?

[Anonymous No.12438468]

>>12438426
There are 2 canonical ways
As a function of n. x : N -> R with x_n = 1/n
This sequence converges to 0
Recursively x: N -> R with x_1 = 3/2 and x_{n+1} = (2x_n + 2)\(x_n +2)
This sequence converges to sqrt{2}
I don't think a list must give a way to be written to exist I a more a ontological maximalist, if you can think about it, it must exist

How do I use latex here? with \$ \$?

[Anonymous No.12438475]

>>12438467
>in mathematics just take the existence of real numbers as an axiom
yes, when you take an analysis course in college, the reals are usually defined axiomatically as any set that satisfies a set of axioms.
https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach
There are other methods, but they are not as rigorous or they waste too much of your time before you could do real analysis.
>Why do people think the reals don't exist?
lack of formal math education and/or sanity. It's a little sad, really.

[Anonymous No.12438476]

>>12438460
No, not all equations have solutions. Yes, the solution of [math]p^2=2[/math] is [math]p=\sqrt(2)[/math]. In fact p can also be brought up as a limit of a sequence of rational numbers, so not even a solution to an equation (this one however I thought you wouldnt really get).

[Anonymous No.12438479]

>zoomers and newfags constantly posting shitty questions shitting up the catalog instead of asking in the stupid questions general

[Anonymous No.12438483]

>>12438468
>As a function of n. x : N -> R with x_n = 1/n
So is this function the real number 0?
What about the function x_n=1/2n. Would that also be 0, even though they're different functions? Seems confusing.
> if you can think about it, it must exist
Very interesting. But how do you know you can actually think about it and that it's not just a fantasy. I could think about the least odd prime p such that there are nonzero natural integers a^p + b^p = c^p, but this doesn't mean that such a p actually exists, does it?

[Anonymous No.12438488]

>>12438475
Reals dont exist though. We construct them from the rationals, which dont exist either. Not even all integers exist. The true God given numbers are the natural numbers. They exist as a Divine axiom. Many animals innately understand the naturals.

[Anonymous No.12438494]

>>12438475
But if the existence is axiomatic, aren't they right to be concerned? I could also define axiomatically some set which later is proved to contain some contradiction, no? Just because you can define some set doesn't mean it actually exists, does it?
>>12438476
Ok but then how do you know that p^2 = 2 has a solution at all?

[Anonymous No.12438496]

>>12438488
take your meds, bro
>Many animals innately understand the naturals.
including humans. Humans have enough brain capacity to come up with more operations than just addition such as subtraction, multiplication, and division, hence the rationals.

[Anonymous No.12438505]

It's a "representative" (I'm not a native english speaker I am not sure of the word) of the equivalence class, so does the example you gave me.
Maybe we should add the satisfiability to my premise.

[Anonymous No.12438509]

>>12438505
Interesting. And what is this equivalence class that it's a representative of?

[Anonymous No.12438513]

>>12438494
>But if the existence is axiomatic, aren't they right to be concerned?
Yes, they are absolutely, but you should bother a mathematician with it. Go ask a philosopher.
>I could also define axiomatically some set which later is proved to contain some contradiction, no?
That is very true. It all comes down to usefulness and consistency. The reals are nice because you can do calculus on them. The complex numbers are even nicer: many algebraic and analytic properties of complex numbers are much better understood than for the reals. If you ask most professional mathematicians, they have the same pragmatic approach to it.
>Ok but then how do you know that p^2 = 2 has a solution at all?
They don't in the rationals. Which just means if you looked through the entire set of the rationals, you won't find a single number there that satisfies it. It does have a solution in the reals or any algebraic extension of the rationals that contains [math] \sqrt{2} [/math]. This is why when we say a root exists, we always specify the set it exists in in "proper" math.

[Anonymous No.12438516]

>>12438371
>>12438509

[Anonymous No.12438528]

>>12438513
>It all comes down to usefulness and consistency
Interesting. Do we actually know the definition of the reals is consistent?
>The reals are nice because you can do calculus on them
Is there anything that breaks down when you try to do calculus on the rationals? For example, when I took derivatives of polynomials in high school I didn't seem to have any need for these real numbers.

[Anonymous No.12438530]

>>12438509
Every convergent sequence is Cauchy, a Cauchy sequence must converge if it is in a complete space. Every metric space acepts an unique completation up to isometries. The completation of the rational numbers are the real numbers

[Anonymous No.12438536]

>>12438494
>Ok but then how do you know that p^2 = 2 has a solution at all?
this p actually exists as a limit of a sequence of rational numbers, a so-called limit point.

[Anonymous No.12438539]

>>12438516
Interesting. It seems like we run into the same question again. I think we both agree that there are infinitely many such equivalent sequences, and we can't simply write down all of the sequences, so how do we actually specify this set? I.e. what actually is this set?

[Anonymous No.12438542]

>>12438528
ah, that's a pretty deep question which I may be wrong on. Gödel's Incompleteness Theorem says you can't prove whether an axiomatic set is consistent or inconsistent within its own framework. This doesn't mean it's always inconsistent, just that it may be. I don't know of any examples of this that pertain specifically to the reals, but maybe other anons know this.

[Anonymous No.12438549]

>>12438536
You can get good aproximations with this sequence
>>12438468
If you acept that the only number that is grater or equal to 0 an less that every non 0 number is 0, you must accept that this sequence converges to something, and that something satisfies the equation
That's the nice argument behind all calculus and analisis

[Anonymous No.12438556]

>>12438530
Do we actually have to construct such a completion or do we just take it as an axiom?

[Anonymous No.12438558]

>>12438528
>Is there anything that breaks down when you try to do calculus on the rationals?
yes. Your limits that contain only rationals may converge to a number that is not a part of the set of rationals. If we only consider rationals, this in a way means they converge into "nothing".

[Anonymous No.12438561]

>>12438556
It's an axiom, but there are a number of ways to construct it. You can use supremums, Cauchy sequences, Dedekind cuts. All of those are equivalent.

[Anonymous No.12438564]

>>12438556
The completion is similar to this
>>12438371

[Anonymous No.12438566]

>>12438549
>You can get good aproximations with this sequence
This sequence has no end, but its end point exists and it's p. Your calculator will give you the best approximation of p it can sustain.

[Anonymous No.12438572]

>>12438566
*limit point sorry, not end point

[Anonymous No.12438573]

>>12438558
But is that really a problem? Not all sequences converge, so what? For example, the sequence -1,1,-1, 1,... doesn't converge, should we also want to force it to converge?
>>12438561
Can you pick one of your favorites and explain it to me please?

[Anonymous No.12438574]

>>12438561
Dedekind cuts are neat because you can construct them with only the ordered geometry and the existence of parallels

[Anonymous No.12438593]

>>12438573
>Not all sequences converge, so what?
Your example is a different kind of divergence. The values oscillate instead of getting closer and closer to a single number. In the example I gave you can have numbers come closer and closer to a single value, but that value won't be a part of the set of rationals, which causes problems. Consider picrelated. It's a sequence in a sense that you can do these fraction towers consecutively and they will bring you closer and closer to pi, which is irrational.

[Anonymous No.12438594]

>>12438572
I think the universe is finite but I also think natural number are infinite, real numbers exists and the cardinality of the real numbers is the cardinality of the power set of the natural nubers wich is extrictly bigger than the cardinality of the naturals

[Anonymous No.12438603]

>>12438573
>Can you pick one of your favorites and explain it to me please?
Supremums and Cauchy sequences were already mentioned in
>>12438394
and
>>12438371
respectively. I'm too lazy to explain Dedekind cuts, but they are really cool conceptually.

[Anonymous No.12438604]

>>12438593
But your sequence doesn't actually converge to any number, at least not in the rationals. Why do we want to force things by jamming new numbers such that they do converge? Is there any actual benefit to this? What can you do by jamming these numbers that you can't do by simply working in the rationals?

[Anonymous No.12438612]

>>12438604
>But your sequence doesn't actually converge to any number, at least not in the rationals
Indeed, that's why we construct the reals.
>Is there any actual benefit to this?
calculus

[Anonymous No.12438615]

>>12438612
What do you need the reals for in calculus? As I mentioned before, I've done some calculus in high school and a lot of it could be done just with the rational numbers.

[Anonymous No.12438623]

>>12438603
Can you explain to me the one you understand best? I still don't understand any of them and would really want someone to explain them for me.

[Anonymous No.12438627]

>>12438615
https://www.youtube.com/watch?v=vV7ZuouUSfs&ab_channel=MichaelPenn
This video talks about that

[Anonymous No.12438639]

>>12438615
>What do you need the reals for in calculus?
to have your Cauchy sequences to converge to something. You need limits to do derivatives, etc. You can't take derivatives if something underlying them are pathological.
>>12438623
If these explanations are too tough for you my best advice would just be to stick with the "plugging the holes" explanation and not think about it too much. Or you can watch some yt. Visualization helps sometimes.

[Anonymous No.12438646]

>>12438627
one of my favorite math youtubers. He can be dry and monotonous at times, but I love his rigor and clear explanations.

[Anonymous No.12438649]

>>12438615
Not him. I think you're borderline trolling at this point. Yes, the reals are a mathematical construction from the rational numbers, a literal ass pull. With the reals you can construct analysis formally, and from analysis you can construct calculus formally by proving each plug-and-chug formula you THOUGHT you were using in high school but actually you had no idea what you were doing.

I can only imagine what happens when you find out about complex numbers, lmao

[Anonymous No.12438652]

>>12438627
Watching the video right now. This is interesting. His example is a function that should not be continuous but is continuous in the rational number. I cannot help but notice that this function would no longer be continuous if you just alter the definition of continuity to say that f:Q->Q is continuous if for any Cauchy sequence (x_n) the sequence (f(x_n)) is also Cauchy. Seems like a simple fix.

[Anonymous No.12438653]

>>12438649
I'd give OP the benefit of the doubt. He sounds legitimately interested and somebody who isn't trolling.

[Anonymous No.12438675]

>>12438652
x+1/n is Cauchy but doesn't work

[Anonymous No.12438687]

>>12438639
>to have your Cauchy sequences to converge to something
Yeah I understand that, I'm just interested in whether this is really necessary.
Limits also sometimes exist in the rational numbers. For example, to take the limit if x_n=1/n you don't need to jam any new numbers.
>Visualization helps sometimes.
Can you expand of this? I've been told I can imagine a real number as a point on the number line but is that really possible? Whenever I draw a point anywhere, when I actually zoom in I see that it covers a whole area, not just one point. I also think that if I can imagine such an infinitely thin point, I can also imagine a point on the number line that's infinitely close to zero but not zero. But that's not a real number is it?
>>12438649
>Yes, the reals are a mathematical construction from the rational numbers, a literal ass pull
Can you please pick your favorite construction and explain how it works? I would love to understand it.
>THOUGHT you were using in high school but actually you had no idea what you were doing.
Why do you say that? I think I had a pretty clear idea of what I was doing. The whole thing was pretty intuitive.

[Anonymous No.12438690]

>>12438675
What do you mean? My point is that if you take any sequence of rational numbers that jump between bigger and smaller than sqrt(2) then the image of this sequence would not be Cauchy, which shows that this function is not continuous under my new definition.

[Anonymous No.12438691]

>>12438687
>Limits also sometimes exist in the rational numbers.
the key word is "sometimes". We want something that will work every time.
>Can you expand of this?
watch yt

[Anonymous No.12438693]

>>12438627
Also looking at his example of the intermediate value theorem and my instinct is that of course it doesn't work, since you no longer plug up the holes. But I don't see that as an issue. Isn't all that matters is that you can approximate the zero, which works for my given definition of a continuous rational funtion?

[Anonymous No.12438707]

>>12438691
>We want something that will work every time.
Again, why? How does it help anything having the sequences converge?
>watch yt
Link to the video please.

[Anonymous No.12438711]

>>12438687
>Can you please pick your favorite construction and explain how it works?
I'll be informal. People on this thread have been talking about sequences. Now if you have time to lose, pick up your calculator. Assume there is no solution to p^2=2, but you wan to find an approximate number k such that k^2 is quite close to two. As you try and try, you will find that you can get arbitrarily close to two. Apparently there is no end to how much closer to 2 you can get. You are actively working with a sequence that is converging to some 'number'. We have no way to see the full value of this number so I will call it [math]\sqrt{2}[/math], right out of my ass. This thing can be done with an infinite number of other irrational numbers, and I take them all with all rationals to make up the reals.

[Anonymous No.12438713]

>>12438707
We will later explore weirder places than the real numbers, the journey can lead us to the hearth of reality

[Anonymous No.12438724]

>>12438707
>Again, why?
to have a non-pathological theory. Otherwise you will lose sanity trying to figure out each time whether or not your theory works or not for a particular case.
>How does it help anything having the sequences converge?
You really need to work on your reading comprehension. I'm saying we want the sequences to converge every time they approach a bounded value, and don't oscillate or jump to infinity, so that calculus will work every time with no exceptions aka my first point.

[Anonymous No.12438727]

>>12438711
Why can't we just say that it doesn't converge to any rational number and leave it at that? Why do we need to jam new irrational numbers into our system?
And what exactly do you mean by a sequence? I get that if I iterate an algorithm like x_n+1= (x_n + 2/x_n)/2 I get a sequence, but I mean what do we mean by a general sequence? Does it need to be given by an algorithm? Or can it have an arbitrary description? Or is it any collection we want, even without necessarily a description?

[Anonymous No.12438731]

>>12438724
>to have a non-pathological theory
And what is pathological with just working over the rationals?
You or some other anon posted a video by Michael Penn and I already explained why I think he's wrong.
>we want the sequences to converge every time they approach a bounded value, and don't oscillate or jump to infinity, so that calculus will work every time with no exceptions aka my first point.
And what about calculus doesn't work exactly if we have such sequence that don't converge?

[Anonymous No.12438733]

>>12438727
man, you really must be trolling
for calculus to work, retard. That's it. We want to do calculus. Doing calculus in the rationals leads to all kinds of problems outlined a hundred times in this thread already.

[Anonymous No.12438736]

>>12438731
>And what is pathological with just working over the rationals?
The fucking examples given to you for instance in the yt video here? >>12438627
really nigga?

[Anonymous No.12438738]

>>12438733
>Doing calculus in the rationals leads to all kinds of problems outlined a hundred times in this thread already.
Like what? Someone posted a video by Michael Penn but I already explained how I think the things he mentioned are not actually problems and how he's overlooking some simple facts.

[Anonymous No.12438740]

>>12438736
I already explained how I think his examples are not really examples of pathology and not troublesome. Which of his examples do you think is particularly troublesome? We can talk about it.

[Anonymous No.12438741]

>>12438738
man, and I thought this is a serious thread...
fuck y'all, I'm outta here

[Anonymous No.12438743]

>>12438741
I don't know about you but I'm being completely serious. Why do you think I'm not?
I explained my objections to his video in these posts:
>>12438652
>>12438690
>>12438693

[Anonymous No.12438744]

>>12438727
We need the reals to make up a number line in which all converging sequences converge to a number in our number line (not to some mysterious thing you can arbitrarily approximate with rationals like sqrt(2)). This is crucial to define everything you have seen in your calculus courses.

A sequence is not necessarily an algorithm. It is (unsurprisingly) any sequence of numbers.

[Anonymous No.12438755]

>>12438743
ok, let's go over the list
>>12438652
was given a counterexample >>12438675
>>12438690
>under my new definition
nobody cares about your definition. There is one definition we care about and it's the one with the limits ruinously defined without using lousy terms like "jump between". For this proper definition, it doesn't work. Believe me, there are many schizos out there who come up with their own "definitions", but all of them fail at one point or another. Unless you have a PhD in analysis, don't even bother.
>>12438693
>approximate the zero
We want to work with it as an element of a set. You can't approximate something that formally doesn't even exist.

[Anonymous No.12438756]

>>12438744
>This is crucial to define everything you have seen in your calculus courses.
Is it? Which particular thing in calculus is it crucial to?
>A sequence is not necessarily an algorithm. It is (unsurprisingly) any sequence of numbers.
Interesting. So can I for example take a sequence that enumerates all real numbers (even though you can't describe it by Cantor's argument)? What if I take the sequence of all expressions in the english language plus some math symbols, then pick out only those expressions which define real numbers, then I let the n'th term of my sequence (x_n) be the n'th digit of the n'th real number in my previous sequence. Is this also a valid sequence?

[Anonymous No.12438759]

>>12438743
I just don't get why are you so reluctant to accept that there are valid reasons out there for us to use the reals and you're insisting that a workaround can be made. Believe me if there were one, we'd be using it right now.

[Anonymous No.12438764]

>>12438755
>was given a counterexample >>12438675
Can you explain how this is a counterexample? What does he even mean by it?
>nobody cares about your definition
The definition is just an obvious fix that makes all the rational functions we want to be continuous be actually continuous in this definition, like in the real case. What my definition does is instead of checking continuity at every point, is to look at Cauchy sequences, which are "spread out" but still get smaller. This gets rid of the irregularities.
>Believe me, there are many schizos out there who come up with their own "definitions", but all of them fail at one point or another.
Are you implying I'm a schizo? Have I been unclear in anything I've said. I would be glad to explain anything that confused you.
>We want to work with it as an element of a set.
Why? How does that help?

[Anonymous No.12438768]

>>12438759
I am not reluctant to accept it, I just want someone to explain to me what those valid reasons to use the reals are, that's all. That's why I made this thread.

[Anonymous No.12438776]

>>12438756
>Which particular thing in calculus is it crucial to?
Everything, from limits to continuity to differentiability to integration. Pick up a real analysis book, as a detailed answer is 300 pages long on average.

>valid sequence
In the reals, any sequence is a valid sequence as long as it is a sequence of real numbers. Same for the rationals. Problem with sequences laking a logic is that you simply dont know what they are going to do after some term. So usually you see sequences defined by algorithms, recursions, or clear logic (like picking from well defined intervals)

[Anonymous No.12438787]

>>12438756
>So can I for example take a sequence that enumerates all real numbers

No you can't, as there is no logic in this sequence. Can you tell me what the third term is? Or even the first?

[Anonymous No.12438790]

>>12438764
>Can you explain how this is a counterexample? What does he even mean by it?
No I can't. It's out of your reach, judging by the things you said so far in this thread. If you're really interested, buy a textbook on analysis and slowly go through it working through the example problems as well. You only get to stuff like sequences on the 3rd month. This whole thing take a semester in college and a thread on a Bhutanese cow-milking forum is neither enough nor a good format for such an explanation if you want something truly formal.
>The definition is just an obvious fix
it's obvious to you because you don't know what you're talking about. You don't even know what a Cauchy sequence.
>Are you implying I'm a schizo?
no, I'm implying that you're ignorant. I admire your curiosity, but you really need to have a good grasp of mathematics to understand this stuff before you make bold statements like "I fixed it, guys"
>Why? How does that help?
various reasons. Defining a derivative of a function as another function is such an example. Again, buy a book on analysis.

[Anonymous No.12438791]

>>12438776
>Pick up a real analysis book, as a detailed answer is 300 pages long on average
A real analysis book explains how to do analysis with the reals, but it does not explain why the reals are necessary for analysis.
>In the reals, any sequence is a valid sequence as long as it is a sequence of real numbers. Same for the rationals
So take my example.
You make the sequence of all expressions in the english language, in lexicographic ordering. Each expression either describes a particular real number or not. I take the subsequence of all the expressions that do describe real numbers. For each n, I let r_n be the real number that this expression describes. Now I let a new real number r to be defined as 0.d_1d_2.... where each d_i is some digit less than 9 which is different from r_n. This seems to be a contradiction, as r is a real number different from any describable real number but it's also describable itself. What's going on here?

[Anonymous No.12438794]

>>12438790
>You don't even know what a Cauchy sequence
Isn't a Cauchy sequence a sequence (x_n) such that for all rational numbers e>0 there is a natural number N such that for all natural numbers n,m > N, we have |x_n - x_m|<e?
>No I can't. It's out of your reach, judging by the things you said so far in this thread
How is it out of my reach? What's wrong with my fix? Just explain it in your own language then, I'll do my best to understand.

[Anonymous No.12438798]

>>12438791
You are clearly trolling because you know Cantor's proof. Fuck outta here. Bye negrito

[Anonymous No.12438802]

>>12438798
>You are clearly trolling because you know Cantor's proof
Yes I saw it in a Vsauce video? Haven't you watched it?
https://www.youtube.com/watch?v=SrU9YDoXE88

[Anonymous No.12438806]

>>12438802
Vsauce, Michael here
Which set has a higher cardinality? The reals or the set of trolls and schizos who deny their existence?

[Anonymous No.12438808]

>>12438806
Why do you keep calling me a schizo? All I am doing is asking simple questions because I want to learn. Ugh.

[Anonymous No.12438810]

>>12438808
>ugh
ask me how I know you're a autogynephiliac transexual

[Anonymous No.12438817]

>>12438810

[Anonymous No.12438824]

ITT: OP thinks he's socrates and retards fall for it
Good job though OP

[Anonymous No.12438865]

>>12438824
The transexual has been called out and stopped answering. He's probably crying

[Anonymous No.12438872]

Let me try to recap the thread so far:
>I asked what the real numbers are and people explained them as things that fill up the holes in the rational numbers.
>I asked why those holes needed to be filled and no one had an answer.
>Someone posted Michael Penn's video on this which has very obvious flaws which I pointed out.
>Nobody so far has explained how what I said was wrong.
>So we're filling these holes, even though nobody can explain why we're doing it.
>I ask how they are filled, what is the actual construction.
>People refer to equivalence classes of Cauchy sequences and Dedekind cuts.
>Nobody wants to properly explain what they mean by the terms used in the definition.
>I ask whether my given examples are also sequences, no response, nobody wants to engage.
>Get accused for being a troll (for asking too many questions?).
I have got to admit, this is starting to seem a little bit suspicious. I'm sure it's just because this thread is only starting but still it seems weird that such a fundamental concept in mathematics is so difficult to approach and explain. I still gladly welcome any anons who want to explain it to me.

[Anonymous No.12438876]

>>12438865
What do you want me to answer?

[Anonymous No.12438897]

This is all highly vexing.

[Anonymous No.12438908]

>>12438872
sneed

[Anonymous No.12439037]

>>12438394
>>12438488
>>12438539
>>12438593
What is a real world app Latino of this.

[Anonymous No.12439052]

>>12438872
The reals are for brainlets

[Anonymous No.12439055]

>>12439052
>>12438872
Forgot link
https://m.youtube.com/watch?feature=emb_title&v=ZRVQIajVdfs

[Anonymous No.12439058]

Looking through the thread, I don't think OP's lack of understanding of the real numbers is the "real" issue here, this post >>12438791 in particular suggests that it's just a symptom of their lack of understanding of mathematical logic. But setting that straight is going to make teaching the fundamentals of analysis look like a cakewalk, so I'm not even going to attempt it ITT. The most I'm willing to offer is a prompt:

>>12438872
What do you make of the theory of a real closed field https://en.wikipedia.org/wiki/Real_closed_field, specifically with regards to its decidability and consistency (which entails, by downward Lowenheim-Skolem, that it has a countable model, such as the algebraic numbers)?

[Anonymous No.12439091]

>>12438872
just a gist:

we can define a real number as an indexed pair, first being a whole number, the second being an infinite sequence of whole numbers from [0,9], such that what you know and love as (pi) is understood as (3, [1,4,1,5,....])

as long as you define it such that 0.999... = 1.000 and so on for other cases (only trailing 9s colliding with a XX000... are an issue), and that addition and stuff are of common sense behavior, you can show that this system is isomorphic to:

the minimal extension of Q that satisfies completeness (the only total ordered field, up to isomorphism)
dedekind cuts (define each real number as a pair of rationals: the ones larger than "it", and the ones smaller than "it")
(i think?) smallest space where cauchy sequences converge

[Anonymous No.12439101]

>>12438872
Best argument for the existence of the irrationals i've seen would be the simple geometric argument.

1. Squares exist
2. Take a 1x1 square
3. You now have a diagonal of √2, by the pythagorean theorem

Geometry can also point towards the existence of numbers like pi. You could argue that pi doesn't exist, it's just a sequence of numbers that get increasingly closer to the ratio between the circumference of a circle and its diameter

However, I personally find the concept of density to make the most sense, as far as this intuition goes. The rational numbers aren't dense; there's only a countable number of them between 0 and 1. If you choose a point randomly in [0,1] on a number line, then you have a 0% chance of choosing a point with finitely many decimal places. You can keep on zooming forever, and it'll never land exactly on one of the ticks. You've just chosen an irrational number.

[Anonymous No.12439118]

>>12439101
>The rational numbers aren't dense
?

[Anonymous No.12439151]

>>12438345
Doesnt exist.

[Anonymous No.12439152]

>>12438345
start with 1
>introduce counting
now you have Natural Numbers (fancy N)
>introduce addition and subtraction
now you have Integers (fancy Z)
>introduce division and multiplication
now you have Rationals (fancy Q)
(division by 0 not defined)

>introduce exponentiation and roots of polynomials
Now you have Algebraic numbers (Fancy Q)
(note, you now have complex numbers due to sqrt(-1))
>go beyond that
anything beyond that is considered a Transcendental Number

Real numbers contain all of the above except for things like i, or things which are derived from functions that aren't properly defined at certain values (for example, what is the log of a negative number? we could define them as their own number set if we wanted to the same way we do with complex numbers, but there's no point and they're obviously not real numbers). So because the real numbers are a commutative ring with multiplication and addition, only certain numbers with certain properties are allowed to get added in the further down you go, and the rest get filtered out.

[Anonymous No.12439684]

>>12438872
hello mr. I don't understand sequences

[Anonymous No.12439709]

>>12438345
its an element of the closure of Q

[Anonymous No.12439711]

>>12438371

[Anonymous No.12439719]

>>12439711
what's your point?

[Anonymous No.12439861]

>>12439719

[Anonymous No.12439884]

>>12439861
this theorem is false.
proof: by inspection.

[Anonymous No.12439924]

>>12439884
based

[Anonymous No.12439926]

>>12439884
can you immagine this quack actually teaches at a university.

[Anonymous No.12440007]

>>12439058
>What do you make of the theory of a real closed field https://en.wikipedia.org/wiki/Real_closed_field, specifically with regards to its decidability and consistency (which entails, by downward Lowenheim-Skolem, that it has a countable model, such as the algebraic numbers)?
That's all fine and good but seems completely useless as in analysis you need the concept of a function which is not covered by this theory.

[Anonymous No.12440011]

>>12439091
>(3, [1,4,1,5,....])
Interesting. Is this what an infinite sequence looks like? To me it looks quite finite. Or are you using "..." to indicate that you haven't finished writing the sequence? If the latter, why do you say that this is pi? Surely this cannot be the definition of pi, as many rational numbers also begin with 3.1415.....

[Anonymous No.12440015]

>>12439091
>and that addition and stuff are of common sense behavior,
Can you please explain how you do addition and multiplication on your system? Thank you.
>>12439101
>3. You now have a diagonal of √2, by the pythagorean theorem
You seem to be presupposing the existence of √2 here, but is this really valid? The Pythagorean theorem can be viewed to talk only about the areas of the squares constructed on the sides, it does not explicitly talk about the length of the sides involved.

[Anonymous No.12440032]

>>12439684
Hello mr. I don't want to explain sequences.
>>12439709
Interesting. And what is the closure of Q?
>>12439926
Interesting. Can you explain to me why you think he is a quack?

[Anonymous No.12440050]

>>12440032
>Hello mr. I don't want to explain sequences.
see >>12437627

[Anonymous No.12440090]

>>12440050
There cannot be such a wall as everything we build is finite.

[Anonymous No.12440098]

>>12440090
>>12440090
so [math]10^{10^{10^{10^{10}}}}[/math] doesn't exist I guess

[Anonymous No.12440114]

>>12440050
In that case, I have found a contradiction in mathematics!
You make the sequence of all expressions in the english language, in lexicographic ordering. Each expression either describes a particular real number or not. I take the subsequence of all the expressions that do describe real numbers. For each n, I let r_n be the real number that this expression describes. Now I let a new real number r to be defined as 0.d_1d_2.... where each d_i is some digit less than 9 which is different from r_n. This seems to be a contradiction, as r is a real number different from any describable real number but it's also describable itself.

[Anonymous No.12440126]

>>12440114
>I have found
sure you have. https://en.wikipedia.org/wiki/Richard%27s_paradox

[Anonymous No.12440129]

>>12440126
It clearly demonstrates that your naive notion of a sequence as an infinite wall of whiteboards with numbers on them is inadequate.

[Anonymous No.12440148]

>>12440129
it demonstrates that the property "x is definable by an expression in the english language" is not well defined. nice try.

[Anonymous No.12440156]

>>12440148
Why wouldn't it be well-defined? Every expression in the english language either unambiguously defines a real number or it doesn't. This is just the law of excluded middle.
For example "The positive number x such that x*x = 2" is an unambiguous definition of a real number while "The number x such that x*x=2" and "asd3i121" are not.

[Anonymous No.12440186]

>>12440156
>a water either is cold or it isn't. this is just the law of excluded middle.

[Anonymous No.12440321]

>>12439884
Bad inspection. The theorem is true.

[Anonymous No.12440359]

what the fuck does it matter if sqrt(2), sqrt(3), sqrt(5), sqrt(-1) and so on exist or not? the point is, these numbers come up in useful mathematical circumstances, so they need to be rigorously defined so we can work with them

[Anonymous No.12440948]

>>12440011
it bijects onto Z; pi's first few decimals corespond onto that

>>12440015
yes the process for determining addition takes countably infinite steps and never halts for "true irrationals", but i would be led to believe that you can "delay" the expansion of the expansion such that the final process which is infinite will compute the final expression fine enough

countably many sets of countably many things is countable anyways....

>>12440359
ultimately what this guy says is true: these definitions exist for us to make sense and work with them

and in any case, there aren't even that many well defined "true transcendental numbers"

[Anonymous No.12441291]

>>12440321
let's ditch the inspections then. can you prove that all real numbers are equal?

[Anonymous No.12441308]

>>12439861
Prove that the abbreviated sequences are Cauchy.

[Anonymous No.12441363]

>>12440032
the closure of Q is R which is the unique ordered field with the least upper bound property

[Anonymous No.12441364]

>>12441308
Ehm, they're Cauchy by definition, alpha and beta are equivalence classes of Cauchy sequences. An arbitrary Cauchy sequence isn't going to look very nice, not in its initial terms.

[Anonymous No.12441391]

>>12441364
Can't be defined as Cauchy unless the elements get arbitrarily close forever.

[Anonymous No.12441417]

>>12441391
You're getting to the crux of the issue. How do you verify this stuff regarding an arbitrary Cauchy sequence? You unravel the sequence for ever and ever... It's infinite work.

[Anonymous No.12441432]

>>12441417
>You're getting to the crux of the issue. How do you verify this stuff regarding an arbitrary Cauchy sequence? You unravel the sequence for ever and ever... It's infinite work.
so

[Anonymous No.12441466]

>>12441432
So??? Can you do infinite work? No, you can't, which is why you probably just assume you can, and call it a day.

[Anonymous No.12441495]

>>12441417
>How do you verify this stuff regarding an arbitrary Cauchy sequence?
you don't. now tell me, where in practice one actually needs to verify whether some arbitrary sequence satisfies the cauchy condition?

[Anonymous No.12441521]

>>12441417
>an arbitrary Cauchy sequence
No such thing. A Cauchy sequence can only be "arbitrary" in the trivial sense of prepending arbitrary terms to a known ending parameter. If an arbitrary sequence simply oscillates wildly, you don't know that it's Cauchy.

[Anonymous No.12441523]

>>12441466
>So??? Can you do infinite work? No, you can't, which is why you probably just assume you can, and call it a day.
but i literally can do infinite work
watch this
1/n-->0

[Anonymous No.12441529]

>>12441523
How is this infinite work?

[Anonymous No.12441532]

>>12441417
This is me,
>>12441308
>>12441391
>>12441521
Not the intervening posts.

[Anonymous No.12441540]

>>12441529
>making half of a step every time is not infinite work

[Anonymous No.12441600]

>>12441540
What are you even talking about?

[Anonymous No.12442635]

>>12438345

I went full autist and answered you question here: https://pastebin.com/xFx5XwFs

Conclusion: A real number is a set of infinite sets of infinite sets of sets of infinite sets of sets of sets that I like.

[Anonymous No.12442761]

>>12441540
>>12441600
its going back to one of the zeno arguments

[Anonymous No.12442767]

>>12442635
Not gonna click that shit, copy the relevant info here.

[Anonymous No.12442780]

>>12442767
>Not gonna click that shit
based

[Anonymous No.12442879]

>>12441417
>>12441521
You give up already? I'm disappointed.

[Anonymous No.12443474]

>>12442879
not that guy, but why is infinite work bad? in any case i believe that its procedure works in countably infinite steps.....

[Anonymous No.12443651]

>>12443474
My posts are about the proposal of an “arbitrary Cauchy sequence” and why that doesn’t make sense other than trivially. I have no interest in “infinite work” and don’t care.

[Anonymous No.12443854]

>>12442635
Incrediblely based

[Anonymous No.12443886]

>>12441364
lol faggot u got owned

[Anonymous No.12445650]

>>12440011
>(3, [1,4,1,5,....])
Yes, the digits come from the next formula (among others).
[eqn]\sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} \;=\; \frac{\pi}{4}.\![/eqn]
You can take as many digits as you want. The most important thing is that you feel satisfied.

[Anonymous No.12447633]

>>12445650
Explain please how to get the 100000th digit from this formula?

[Anonymous No.12447880]

>>12447633
add the first 10^8 terms or so, look at 100000th digit

[Anonymous No.12447885]

>>12447880
>add the first 10^8 terms or so, look at 100000th digit
How do you know this will give me the 100000th digit of pi?

[Anonymous No.12447913]

>>12447633
why should anyone care what digit occupies the 1/10^99999 place of the base 10 representation of pi? if you really want to know, either do the calculation yourself or google it retard

[Anonymous No.12447920]

>>12447913
I'm only asking because he claimed
>Yes, the digits come from the next formula (among others).
I'm asking how by asking to provide an example of how to get 100000th digit.
>either do the calculation yourself or google it retard
How do I do infinitely many procedures myself?

[Anonymous No.12447923]

>>12447885
the k-th digit stabilizes for n>k large enough. finding explicit bounds for the case k=100000 is left as an exercise for you.

[Anonymous No.12447942]

>>12447920
>the digits come from the next formula
do you disagree that the digits of pi can be calculated to arbitrary length in any base by adding 1/3 1/35 1/99 etc... as per that formula?
>How do I do infinitely many procedures myself?
who asked you to do that?

[Anonymous No.12448431]

>>12447923
>the k-th digit stabilizes for n>k large enough
Prove it.

[Anonymous No.12448433]

>>12447942
>do you disagree that the digits of pi can be calculated to arbitrary length in any base by adding 1/3 1/35 1/99 etc... as per that formula?
I neither agree nor disagree. Do you agree? If so, can you tell me how to do that (using that formula)?

[Anonymous No.12448572]

>>12447920
>How to get 100000th digit
>How do I do infinitely many procedures myself?
What you asked is clearly a finite procedure

[Anonymous No.12448580]

>>12448433
how to do what? calculate the digits to an arbitrary length?

[Anonymous No.12449111]

>>12448431
I told you it's left as an exercise for you

[Anonymous No.12449124]

>>12449111
I give up. You claimed it can be done from that formula, now show us how.

[Anonymous No.12449136]

>>12449124
Show you how to add?

[Anonymous No.12449146]

>>12449136
Add what?

[Anonymous No.12449162]

>>12449146
I'm not the same guy, but it seems like you're asking how to add fractions?

[Anonymous No.12449171]

>>12449162
No, I'm asking how to get the n'th digit from that formula. I know how to add fractions.

[Anonymous No.12449274]

>>12449171
Oh, sure, convert to 8/((4k-3)(4k-1)) and add the first 10^n terms to get the first n digits in base 10. Lim sup on the sum of terms 10^(n-1)+1 to 10^n is 9/20(10^-n) so lim sup of all remaining terms is 1/2(10^-n). So unless digit n+1 is a 9 followed by a 5 or greater, you're good. If it is, and you care enough, go on to the next digit. If you get stuck in a hypothetical wormhole of 9s, ask yourself if you care whether the nth digit is (X)(999...) or (X+1)(000...) and to what extent it even matters.

[Anonymous No.12449501]

>>12449274
it's guaranteed that 9's will end after a finite amount of steps. there's no problem.

[Anonymous No.12449525]

>>12449274
I dont believe what you are saying is possible to do. Can you use your given method to get the 1000th digit?

[Anonymous No.12449532]

>>12449525
sure, it's 9

[Anonymous No.12449571]

>>12449525
I gave exact bounds for how and why that formula gets you to the nth digit. If you want to practice adding fractions for a long time, go for it. I bet you give up before you get there.

[Anonymous No.12449614]

>>12449501
If the limit is π or any other value proven to be irrational, sure. If you fuck up and try running this on the reciprocal triangular numbers, for example, you get stuck in a 9s wormhole lol.

[Anonymous No.12449618]

>>12438345
24 and 37, now those are some fine ass numbers.
I hate 7 and 91, they're not real numbers.

[Anonymous No.12449723]

>>12438345
This

[Anonymous No.12449725]

>>12449618
>91
>not a real number
Fuck you. Eat shit, you fucking braindead retard. I'd knock your fucking lights out if you said that to my face, bitch

[Anonymous No.12450349]

[Anonymous No.12451779]

>>12438345
A real number is an element of the set of real numbers, and no I'm not joking or being cheeky.

[Anonymous No.12451783]

>>12438394
better explained than my professor lmao