/sci/ - Science & Math

do people still think trump is going to win lol
you can use math to make the numbers mean anything you want

>>12370361
https://www.youtube.com/watch?v=H84VyZCOCew

wrong link.

oops.

>>12370361
or course he's going to eventually win. once the case makes it to the supreme court and eventually the house.

>>12370374
cope

>>12370345
Is this guy full of shit? Mathematician I mean obviously

>>12370595
No, he's right about everything. (t. math phd)

tl;dr?

>>12370789
Waste of time, dont watch

>>12370789
Real numbers are fake because there's no way to actually do arithmetic on them to get results like you can with natural numbers or rational numbers.
They rely on weak imprecise definitions, most often what a set is, so neither Dedekind, nor Cauchy nor other supposed definitions of real numbers actually work.

>Waste of time, dont watch

>>12370374
Its already being dismissed by courts everywhere.

>>12370796
(!) This claim about real numbers being fake cannot be disputed.

>>12370796
>>12370820
>They rely on weak imprecise definitions
which rational number is x?

>>12370369

>>12370826
sqrt(x^2)

>>12370826
You seem to be implying someone here says all definable numbers are rational

>>12370833
based retard
>>12370839
if irrational numbers exist precisely their union with rationals is the real numbers. that is quite literally the definition of real numbers, >>>/pol/

>>12370798
please express x as a ration of integers >>12370826

>>12370826
find me an object in this universe, or make one, that has these perfect dimentions. There isn't one and you can't make one, meaning x on a real object is finite and not irrational.

mhmm

>>12370850
how can you then talk about arithmetic over the natural numbers when cutting an apple in half always leaves shit on the knive therefore by your reasoning 4/2 =/= 2, >>>/pol/

sorry i dont have autism so i cant sympathize with anything he complains about

>real numbers are not concrete and programmable in a literal sense
>therefore all math is fake and useless

The dude is better educated than I am, but he's gotten too hung up on this minor philosophical observation that no one is going to seriously argue with him about

>>12370850
you would think that mathematicians, of all people, would be okay with abstractions

>>12370850
ok forget about geometry. give me a rational number solution for x*x = 2

>>12370855
this is really situational. if you have 4 sticks and you divide by 2, you get 2 sticks if you discard the other two. If you cut an apple "in half" (something which is literally impossible), you can get two pieces which one is slightly over 2 and one which is slightly less than 2.

>>12370843
The definition is a "ordered field with least-upper-bound property".
Pointing at the single distance [math]\sqrt 2[/math] that fits somewhere between rationals is not an argument that [math]{\mathbb R}[/math] is a good idea.

>>12370865
"it doesn't exist" is a shit answer, btw. Pretend I am an engineer who depends on finding this value. What's the value, bergertards?

>>12370870
you're gonna get a value very close to 1.41421
you can continue to get more values after the decimal point to your liking. And yes, it doesn't exist.

>>12370857
His position is a bit more extreme, in that he rejects both formalism and infinite sets.

>>12370868
>The definition is a "ordered field with least-upper-bound property".
uh, no its not. it's the disjoint union of Q and I
>>12370870
>Pretend I am an engineer who depends on finding this value
why would i? you claim there is a finite rational number that equals the distance

>>12370857
>The dude is better educated than I am
he's a min-maxxed retard schizo

>>12370345
Can someone explain to me what he means by "fake" and "not existing"?
I am not in the mood to sift through countless hours of youtube ramblings.
If not, that is ok too.

>>12370911
he's just being polite. He literally means that reals are fake AND gay

>>12370911
fake = not existing = I dont personally like it

>>12370911
In praxis, he's saying one should spend time studying unrealizable theories.

On another level, he's some sort of finitist Platonist - which is where his motivation comes from.

>>12370911
https://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf

>can't write down = not real

>>12371033
Interesting, while it seems reasonable to define "existence" in the way he did, his claim that anything to do with reals is not mathematics because of that definition sounds extremely arbitrary to me.

>>12370848
>>12370826
Euclid already proved more than 2000 years ago that there's no rational number that squares to 2.
>>12370843
>that is quite literally the definition of real numbers
No it's not, retard. Talking about "irrational" numbers doesn't make sense if you haven't defined what real numbers are. You define irrational numbers as the real numbers that are not rational, not the other way around.
>>12370854
What's wrong with this?
>>12370856
>implying he has autism
>>12370857
Please tell me what you understand a "real number" to be.
>>12370868
Also Wildberger points out in many of his videos that you can actually make sense of sqrt(2) without resorting to the real numbers by simply defining field extensions of the rationals (all finitistic).

>>12370345
Non-american (so you know I'm unbiased) mathematician here. It is absolutely obvious that this election was fraudulent. I've run the numbers and the probability of getting such a voting distribution less than 0.000000000000000001% using a naive model, and using a model fed with past election data the probability is less than 0.00000000000000000000000000000001%.

>>12371074
Please do tell why you think "real numbers" are a sensible thing to talk about. How would you construct the real numbers?

>>12371078
>Talking about "irrational" numbers doesn't make sense if you haven't defined what real numbers are
no you double nigger retard, irrational numbers are literally not rational that's it. the same way complex numbers extend in a sense the real numbers it's the same for reals extending the rationals

>>12371087
I think they are sensible because they are fun to play around with and give rise to interesting consequences. As for construction, it is not hugely important to me. Dedekind cuts are how was taught to understand them.

>>12371092
>I think they are sensible because they are fun to play around with and give rise to interesting consequences
My imaginary gf who lives in Canada is interesting and fun to play around with but that doesn't make her real.
> As for construction, it is not hugely important to me
So whether or not the central object in all of mathematics actually makes sense is of no importance to you? Weird.
>Dedekind cuts are how was taught to understand them.
This is a fake construction that doesn't work. If you think it does, please do tell what is meant by a "subset of the rationals".

>>12371091
Of course irrational numbers are not rational. However, you can't define the real numbers as the union of the rational numbers and irrational numbers because you don't know WHAT it is that is being irrational in "irrational numbers".
According to you, it seems like 8i is a real number because it's not a rational number. Clearly that is ridiculous.

>>12371105
>Weird
I could say the same about your idea of "reality" and "fakeness". It's pointless arguing about this because we have obviously very different conceptions of what maths should be like. And I don't think there is any objectivity in deciding that.

can one of you tldr this for a brainlet

>>12371142
> It's pointless arguing about this because we have obviously very different conceptions of what maths should be like
Hmm... I think maths should be rigorous and precise and you think maths should be ????
It's fine to think about vague and nebulous concepts, to see what kind of consequences they entail, mathematicians have been doing that for a long time in a lot of areas of math, just don't pretend you're being rigorous and that the issue of real numbers is settled. Nothing about the real numbers is rigorous.
>>12371143
Mathematicians really wanted an object to exist so they came up with fake definitions that don't work and started pretending like the issue is settled (they got so desperate that they even decided to call this object "real numbers" to further insist on the point that THEY'RE NOT FAKE GUYS I SWEAR). Wildberger is simply pointing this out.

>>12371158
Oh I see
I think we more or less just make up languages for something that we will never fully understand, and those languages just let is do different things.
The empiricism seems like the fallacy.

>>12371158
Also how does this apply to the real world?
When I have an object, I think of it as having 1 of that then, then when I acquire another, I have 2.
What's the issue?

>>12371173
There's no issue with natural numbers like 1,2,3,. We're talking about fake numbers like pi.
>>12371167
This is not about empiricism.

>>12371158
>just don't pretend you're being rigorous and that the issue of real numbers is settled
I never did and you are yet again using words as if they had a universally agreed-upon definition ("rigorous" in this case).

dunno about him but the math we know and love so far has brought us to the moon.

>>12370808
Not necessarily a bad thing as the cases need to be brought before lower courts first before progressing to SC.

>>12371301
It also exterminated 10 million jews

>>12371301
the math that we used to go to the moon was precisely the math that Wildberger is talking about, applied computational math. Empirical math.
the math that we have right now and use as the framework for cutting edge mathematical research is the opposite, it's a platonic realm circlejerk of extreme rationalism that has zero attachment to reality nor will ever be related to reality in any capacity, because the moment you even enter into the basics of set theory you've already severed the tie to empirical reality forever because this is literally the point of set theory, to construct different sets that do not actually exist. Our world and our computers can run completely fine on the math that was available in the 17th century and everything since has been a pointless descent into delusional platonism that never actually bore fruit

>>12371461
you have no idea what you’re talking about. There is no empirical math you fucking pseud faggot.

>>12371461
lmao, dumb empiricist thinking he can live without the forms.

lol at the BLOOMS comment in the video

>>12371301
that just means it had good parts - you can easily argue we would be further without emphasis on some of its sketchy gimmicks

>>12371475
the fact that there's a concise theory telling me it's best to bet p=1/8 on average, on the probability that 3 coinflips all turn had surely has an empirical touch to it. Sure, it's not like we are forced to adopt the math axioms and theories and models that allow for making good predictions, but there's a part of math that can be tied to it quite usefully - and some people may make a case to put more focus on that.

>>12370808
what is your source? the same media that called the election for biden and refused to let anybody mention the possibility of a legal challenge?
we won't know anything until the electoral college

>>12371808
I'm gonna screencap your comment and post it on 4chan when Biden is inaugurated.

>>12370345
God bless Sidney Powell

>>12370369
"Mathematics is fake. I made it up."
-Thales

Modern mathematics is actually about fake things, there is no controversy on the use of this "fake things" approach. Is this so hard to understand?

>>12371461
>zero attachment to reality
i.e. math. Always has been.

>>12373718
>>12371461
The issue with real numbers is not simply because you can't actually perform any of the operations of arithmetic with them in general (although that's an important point), rather it's that all of the supposed "definitions" of the real numbers are vague and imprecise, which should be unacceptable in mathematics. In mathematics you should be precise with what you mean when you're talking about it, it's called being rigorous.
>>12371475
There is math that is computational, for example most of the theorems in finite combinatorics can be verified and exploited using concrete calculations with a computer. Same with a lot of number theory, algebraic number theory, group theory etc.
Real numbers are non computational. There is no algorithm for adding two real numbers or multiplying them. When you do computations with real numbers there is no actual way to find an answer, you just pretend there is an answer in some platonic realm that you possibly will never get access to and move on.
But the issue with real numbers is not even that it's not computational. The issue is that all "definitions" of real numbers rely on vague, undefined concepts and are not precise at all. By extension, none of the mathematics relying on real numbers is rigorous, and mathematicians need to stop pretending they are.

>>12373714
The issue is not that the objects are fake. You can't find polynomials, finite extensions of rational numbers or graphs in nature, they are not real (fake). However, the difference between these aforementioned concepts and real numbers is that the former have a precise definition and can be reasoned about rigorously. Nobody has been able to define what they mean by a "real number". It's a vague, nebulous concept and should not be considered mathematics.

>>12373832
>Nobody has been able to define what they mean by a "real number".
a limit of a cauchy sequence of rational numbers

>>12373841
What do you mean by a limit of a cauchy sequence of rational numbers?

>>12373843
is there a word you don't understand?

>>12373847
Yes, I want you to explain the word "limit" and a "sequence of rational numbers".
What kind of limits are we talking about here? We know that there are some rational sequences (which most people agree are sequences like the recursive sequence a_0=1, a_n+1= (a_n + 2/a_n)/2 ) that don't have a rational limit, so I assume you're looking at limits in some bigger set. What set is it?

>>12373854
>limit
https://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_definition
>sequence of rational numbers
https://en.wikipedia.org/wiki/Sequence#Definition
>What set is it?
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences

>>12373871
>We call {\displaystyle x}x the limit of the sequence {\displaystyle (x_{n})}(x_{n}) if the following condition holds:
>For each real number {\displaystyle \epsilon >0}
>real number
Circular definition, relies on the concept of the real numbers being already constructed.
>In this article, a sequence is formally defined as a function whose domain is an interval of integers.
This is not a precise definition because it does not explain what is and what is not a function.
>https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences
Ok so that's the actual definition. Do you realize how saying "limits of rational sequences" don't make sense unless you have a bigger set that you're talking about?

>>12373896
how come you're voicing opinions on real numbers when you don't even know how they're defined/constructed?

>>12373919
I obviously know how they're defined, probably more "definitions" than you do. The reason I asked is to explain how your preferred definition is actually fake and unrigorous.
Going through all of the definitions and debunking every single one of them would take too much time.
On the other hand, you don't even seem to understand the definition of reals yourself. At first you gave a completely wrong and meaningless (self-referential) definition even by modern mathematicians standard, then when called out you resorted to linking wikipedia pages which would be completely unnecessary if you understood what you were talking about and were capable of explaining it to me.

You're all a bunch of faggot polemics. Please commit mass suicide.

>>12373927
>Going through all of the definitions and debunking every single one of them would take too much time.
no need to debunk all of them, just the cauchy sequences one will suffice

>>12373934
You're way over your head, retard. Go to another thread.

>>12373937
Still waiting for your answer.
Explain PRECISELY what you mean by a sequence of rational numbers, i.e. what is a function from the natural numbers to the rationals. How do you define it?

>>12371081
What do you mean by "voting distribution"?

>>12373938
No, suck my cock you stupid nigger.

>>12373940
I think there's no point in discussing if you don't even know how functions are defined, man...

>>12373960
It seems like a simple question. Without a clear notion of what a function is and is not the definition is not rigorous.
For example, let f be defined as for any n, I let f(n) to be 1 if n codes a program in C# that halts and 0 otherwise. Is f a valid function? How about g(n)=1 if n codes a sentence in ZFC that's provable and 0 otherwise. Is g a function? What if I let h(n) be the the number of hairs on the n'th person in USA ordered by the national id. Is h a function?

>>12373985
there's no issue with defining functions in ZFC. subset of cartesian product such that blah blah

>>12370345
Absolute schizo tier take.
1) He talks about computers being unable to compute real rumbers, he's right, its not bc the real numbers are fake or whatever he claims, its because our computers are discrete. You can could design a mechanical computer in the abstract that could do any of these operations with absolute precision. We could define the angle of some gear to be the input to some function, then the computer rotates it a certain number of times then halts with the exact answer, however, it would unmeasurable in the real world since our measurement tools have finite precision.

Also, he disproves his own point. He asserts that arithmetic can be done on rational numbers, however, you can get irrational numbers from that. For example we can define 2^(5/2), which is irrational number. So his own argument is incoherent.

>>12374017
ZFC does not help you define a subset at all. A subset in set theory is just a SET such that every element of that set is also an element of the larger set. ZFC says very little about what is or is not a set. It leaves the notion of a set completely undefined. Perhaps you are able to explain to me what set is?

>>12374050
>It leaves the notion of a set completely undefined
Yes, for 'set' is a primitive notion in this first-order formal system.

>>12370865
Fraction 2/2 * 2/2 = 4/2 = 2 so 1 * 1 = 2
:)

>>12374066
I'm just playing with the last part if you don't get that.

>>12371118
fair enough

>>12374050
it's time for you to define something yourself. just so we can see what you consider a rigorous definition.

>>12374064
It's a vague, undefined notion and by extension so is your notion of a "function". There's nothing wrong a priori with having primitive notions, the natural numbers are largely a primitive notion for most people. The difference is that the notion of a set is too vague to be meaningful, people disagree with what is a set or what isn't, it's impossible in general to compare two sets or to answer even the simplest questions about them.
With natural numbers, however, there is no disagreement. Everyone understands and agrees that 10 is a natural number, which is the number followed by 9. It's a clear and precise system and you can reason and compute with them. There are many ways to construct them in a precise way, for example as a string of strokes, as a string of symbols in {0,1,2,3,4,5,6,7,8,9} and so on. Same with rational numbers.
With natural numbers you give a precise account of what they are and then can unambiguously and rigorously manipulate them and calculate with them.
In contrast, nobody knows what a set is. People like Cantor tried to define what a set is but all such attempts failed miserably with logical contradictions.

>>12374095
A natural number is a string of "|"s.
Example:
|||| is the natural number called 4.
|||||| is the natural number 5.
This is a clear, precise and unambiguous definition. There are many other equivalent definitions.

>>12374146
>natural numbers are well defined because people agree on what they are
people also agree on what real numbers are, anon

>>12374180
Wrong. Set theorists disagree on the cardinality of the continuum which is precisely because they disagree on what a real number is. Some believe it's aleph_1, some believe it's aleph_2, some have even changed their mind on what it is during their lifetime (Woodin).
If real numbers were properly defined, there would be no disagreement.

>>12374018
This. I think the issue is that people don't get the idea of pure abstraction.

>>12371033
The rational sequence cavil described here (a trillion billion 2/3 followed by an arbitrary convergence to -17) only matters in the middle. It has nothing to do with the ε𝛿 description of real numbers.

If you say it ends at -17, it's -17
If you say it ends at 2/3 it's 2/3
Who cares what a sequence does in the middle if you already know the end?

>>12374018
>You can could design a mechanical computer in the abstract that could do any of these operations with absolute precision
Except you can't. Arithmetic with real numbers is not computable even in the abstract.
>He asserts that arithmetic can be done on rational numbers, however, you can get irrational numbers from that
No you can't, retard. Arithmetic with rational numbers only gives you rational numbers.
>For example we can define 2^(5/2)
To define it you need to go beyond rational numbers. In the arithmetic of rational numbers, you can only exponentiate by integers, so 2^(5/2) is undefined.
> So his own argument is incoherent.
It's coherent, you're just too stupid to understand it.
>>12374509
Wildberger and finitists in general have no issue with pure abstraction. Graphs, natural numbers, rational numbers, polynomials are pure abstractions and Wildberger is comfortable with them.
The issue comes when you start using vague, nebulous concepts without explaining what you mean by them. That's not pure abstraction, that's tomfoolery and does not deserve to be called mathematics.

>>12370866
And when you pick up one of the sticks, some of its molecules rub off onto your hand. Your low iq can get around the idea that math defines relations, not just quantities.

>>12374516
>cavil
?
>It has nothing to do with the ε𝛿 description of real numbers.
You mean Cauchy sequence description? There's no 𝛿 in it (i.e. there's only one small variable in the definition).
>Who cares what a sequence does in the middle if you already know the end?
The sequence doesn't end by definition. It's ongoing.
His point is that no matter the number of terms you know in the sequence, that will never be enough to determine whether it's Cauchy or what number it converges to.

>>12370850
>what is ramp?
Hurr

>>12374549
>?
https://en.wiktionary.org/wiki/cavil
>only one small variable
If you know the limit, just reduce that sequence to the low water mark of its elements. If you don't know the limit, that sequence has nothing to do with a "real number."
>It's ongoing
And you either do or don't know if it "converges."

>that will never be enough to determine whether it's Cauchy or what number it converges to
Explain this more. I think your cavil is circular.

>>12374583
>If you don't know the limit, that sequence has nothing to do with a "real number."
Except it does. That's that whole point of real numbers: to pretend all sequences that seem like they have a limit actually do, and the limit is the sequence itself. Real numbers are there so that every Cauchy sequence has a limit, even if we can't find it.
>And you either do or don't know if it "converges."
That's the point, it's impossible to know whether a general given sequence converges or is Cauchy because that would require an infinite amount of information and our minds are finite.
>Explain this more. I think your cavil is circular.
The definition of a Cauchy sequence: a sequence of rational numbers (q_n) is Cauchy if for every rational number e>0 there is a natural number N such that for all numbers n,m >=N we have |q_n - q_m|<e.
Checking this condition requires infinite amount of work:
for any given e>0 and N need to check infinitely many terms to find that they're all close together and also there are infinitely many e's to check.

>>12374601
Your argument seems to be that sequences which are indeterminately (may or may not be ) Cauchy sequences somehow impinge on sequences which are determinately (are) Cauchy sequences.
I don't think anyone disagrees with that...

>>12374066
Are you saying you genuinely believe that 2/2*2/2=4/2?

>>12374637
That's not my argument. In the post above I merely explained what Wildberger meant by the passage, which clearly confused you.

>>12374660
No, it clearly confused you, since you don't know how to defend it.

>>12370863
My foot up your ass is an abstraction and mathematicians should therefore be okay with that

>>12374693
Lol. The passage is not an argument, it's just an illustration of just how unintuitive, weird and totally disconnected from real life the notion of a Cauchy sequence is.

>>12374719
And I explained the flaw in the illustration. Feel free to explain the flaw in my explanation.

>>12374755
What's the flaw? Explain again please, this time make it coherent.

>>12374762
>>12374583

>>12374785
>>12374601

>>12374790
>>12374637

>>12374802
>>12374660

>>12374814
>>12374693

>>12374816
>>12374719

>>12374821
>>12374755
And now we’re back around.
You’ve proposed an illustration you read online. I’ve explained the flaw and asked if you can clarify. Instead of clarifying, all you can do is redirect me back to the flawed illustration you read online.
Lol.

>real numbers not exist
>approximations by rationals is totally fine tho
>mfw real numbers are quite literally "approximations by rationals"

>>12374866
Approximations are rational numbers. Real numbers are not rational numbers, retard.
>>12374862
>I’ve explained the flaw
Except you didn't, retard. You asked some elementary questions about the construction of real numbers and I've answered them. What more do you want?

>>12374882
>retard
You already lose here lmao. Try again

>>12374882
>Approximations are rational numbers. Real numbers are not rational numbers, retard.
what the fuck are you talking about nigger

>>12374892
Real numbers are "approximations by rationals", that's plainly false. If that were true, wildberger would have no issue with it. The closest actually true description would be "approximations by a sequences of rationals" which doesn't work because the notion of a "sequence" is always left undefined.

>>12370345
Does anyone have the link where he explains how to do calculus or trig without infinities?

Note that the protagonist in this thread doesn’t know how to address the simple flaw in the illustration they’ve proposed. A sequence that may or may not be a Cauchy sequence has nothing to do with a sequence that is a Cauchy sequence.

>>12374960
>A sequence that may or may not be a Cauchy sequence has nothing to do with a sequence that is a Cauchy sequence.
Yeah, and? How is that a flaw in the illustration, retard?

>>12374967
I read the pdf. Did you?

>>12374984
>Did you?
Yes.

>>12374991
Lol.

>>12374960
>An apple that may or may not be red has nothing to do with an apple that is red.
I can't parse the intention behind your sentence

>>12375004
>Still waiting for you to point out a flaw.

>>12375019
The “illustration” illustrates a sequence that may or may not be a Cauchy sequence and points out that it may or may not be a Cauchy sequence.

This has nothing to do with real numbers, which aren’t “may or may not.”

>>12375059
He very clearly states that the sequence DOES converge, so it follows that it IS a Cauchy sequence.

>>12370345
based

>>12375104
Yes, because he defines it as a sequence that converges to -17.
The argument he makes is that it’s undecided until he decides.
So?

The fact that an unknown sequence of rational numbers isn’t a Cauchy sequence unless it is a Cauchy sequence has nothing to do with how well Cauchy sequences do or don’t describe the continuum. It’s a non sequitur.

>>12375143
>The fact that an unknown sequence of rational numbers isn’t a Cauchy sequence unless it is a Cauchy sequence
This is a tautology and has nothing to do with Wildberge's example.
>has nothing to do with how well Cauchy sequences do or don’t describe the continuum. It’s a non sequitur.
Obviously.

>>12375159
That’s literally the example. I don’t think you’ve read the whole pdf.

He describes a Cauchy sequence of rational numbers which is starts off at exactly 2/3 then converges you -17. He writes that we have no way of knowing what the sequence will converge to until he tells us what the sequence will converge to. None of that is problematic at all. His sequence isn’t a Cauchy sequence until he defines it to be a Cauchy sequence. There’s nothing confusing or even slightly inappropriate at all.

>>12375159
That’s literally the example. I don’t think you’ve read the whole pdf.

He describes a Cauchy sequence of rational numbers which starts off at exactly 2/3 then converges to -17. He writes that we have no way of knowing what the sequence will converge to until he tells us what the sequence will converge to. None of that is problematic in the least. His sequence isn’t a Cauchy sequence until he defines it to be a Cauchy sequence. There’s nothing confusing or even slightly inappropriate at all.

>>12373824
Turns out mathematics isn’t all encompassing as a language

>>12371461
>implying math has anything to do with empirical reality

>>12370345
como siemrpe los hilos retrasaos vienen de españa... vuelvete a youtube a comentar en los videos de estos gilipollas

lmao imagine not believing in real numbers it literally says "real" right in the name

>>12370850
cope

>>12376211
this, desu

>>12370826
Oh no no no no no. We got too cocky, math bros.

>>12370595
A mathematician using math without the Reals is like a guy who can type faster with one hand than everyone else can with two: fun and interesting, I guess, but what idiot would use this to actually solve a problem?
Not allowing Reals leads to problems that are "unsolvable" in Wildeberg-istan, but are trivial with the Reals. For example, finding x in x^2=2. No solution exists in the Rationals, but one DOES exist in the Reals.

>>12379062
A solution exists in the finite field extension Q[x]/(x^2 - 2), as well as in computable numbers (which are uncountable and complete BTW).

>>12371177
You being too retarded to understand something doesn't make it fake.

>>12371033
What a retard, he should just take the computable set as an example and build the set of "practical" numbers, then define them for numbers that you can write down in this universe.

>>12371402
Indeed. Thank you for accepting that the Reals are real.

>>12379075
I probably understand the "constructions" of the reals numbers better than anyone else ITT, which is why I realize (as does Wildberger) that they're complete bullshit and don't work.
I would also be willing to bet that you don't even know any "constructions" beyond the Dedekind cuts and Cauchy sequences.

>>12379077
>should just take the computable set as an example
Example of what?
>and build the set of "practical" numbers
Explain.

>>12374531
>No you can't, retard. Arithmetic with rational numbers only gives you rational numbers.
Oh rly nao? What's 2^(1/2)?
2 and 1/2 are rational numbers, and exponentiation is an arithmetic operation. Thus, by YOUR argument, sqrt(2) is a rational number. What is the rational expression of it? Don't make excuses so that you don't have to answer. Fucking answer it.

>>12374601
>Checking this condition requires infinite amount of work
No it fucking doesn't, you fucking retard. Math undergrads literally solve these Cauchy sequences for homework every school night. Just because YOU are too stupid to use any technique other than brute force, doesn't mean everyone else is.

>>12379107
The arithmetic operations in the rational numbers are +, -, *, /. You are only allowed to exponentiate by integer exponents.
>What's 2^(1/2)
Explain what you mean by 2^(1/2). In rational numbers this is undefined. In the ring Z/2Z this is simply 0. In Z/7Z this is +-3. What number system are you referring to here?

>>12379094
Computable numbers are numbers that can be computed by a finite algorithm. An uncomputable number can't be calculated by a turing machine, as an example the probability that a randomly generated program will terminate.

If he is so angry with numbers that are too big, he can just define a finite set that has only numbers that, let's say can be represented by a binary number with X bits.

>>12379126
>will terminate
Will halt is a better term. I don't know if there are other famous uncomputable numbers, this one in special has a name I do not record.

>>12379118
>No it fucking doesn't, you fucking retard.
In general, it does. A lot of open problems in mathematics could be reformulated as checking whether a certain rational sequence is Cauchy or not.
>Math undergrads literally solve these Cauchy sequences for homework every school night.
This is because the sequences they deal with are always very contrived, artificial and completely unrepresentative of what a general sequence looks like (assuming it's meaningful to talk about a general sequence, which is doubtful).

>>12379126
>An uncomputable number can't be calculated by a turing machine
There are no uncomputable numbers.
>as an example the probability that a randomly generated program will terminate
Explain, without simply linking to the wikipedia article, what you mean by this and why you think this is a valid, well-defined number.
>If he is so angry with numbers that are too big, he can just define a finite set that has only numbers that, let's say can be represented by a binary number with X bits.
Your thinking is too simplistic here. Obviously that wouldn't work because for any such rigid condition either one could find an example of an obviously valid number not in his set or something in his set that's not a number.
Wildberger in general objects to set theory and would probably see no use in such a futile construction.

>>12379083
>I probably understand the "constructions" of the reals numbers better than anyone else ITT, which is why I realize (as does Wildberger) that they're complete bullshit and don't work.
What do you mean by "don't work"? Are there math problems that Mr. Wild can solve that math using the Reals can't? Is there a contradiction in the axioms of the Reals (and I mean an ACTUAL contradiction, not the typical "too many digits after a decimal point scares me" argument from incredulity garbage that retards like you use)?

>>12379125
>You are only allowed to exponentiate by integer exponents.
How fucking convenient...

>Explain what you mean by 2^(1/2).
It's the square root of 2, written in exponential form. Stop pretending to be retarded.

>In rational numbers this is undefined.
Which is a FLAW in the rationals that are fixed by the Reals.

>>12379144
>There are no uncomputable numbers.
Ok...
>Explain, without simply linking to the wikipedia article, what you mean by this and why you think this is a valid, well-defined number.
There's this thing in computer science called the halting problem. Computer programs have this property, in a turing machine they can either run for some time and then halt, or they can run forever, not halting. All programs in a classical turing machine fall in one of those two categories, no exception. A turing machine _cannot_ determine if a certain program halts. That is something well known for a century now, and the proof is very simple. Remember this part.

Imagine you generate all possible programs, absolutely all of them. Then we ask God to tell us which programs halt. Then we divide the number of programs that halt by the total number of programs. God can do that, he is omniscient. It is very clear that if all programas halt, then our number is 1, if none halts, our number is zero. Anything else (which we know is the case) is a number inbetween 1 and 0.

But remember, we asked God for telling us which programs halt, as I've explained before, a turing machine cannot determine that. So if it can't do the first step, it can't do the rest, therefore it can't calculate that magical probability.

>Your thinking is too simplistic here
Well fuck, the your quick explanation was too simplistic then.

>>12379147
>What do you mean by "don't work"?
They don't work logically because all of them ultimately rely on vague, undefined notions such as a function or a set. Not vague as in a primitive notion that although undefined, people understand what it means, vague as in nobody actually knows what is meant by those words. The results of Cohen and Godel w.r.t. Continuum Hypothesis, as well as the subsequent failed attempts to resolve CH by appealing to large cardinals are a tangible mathematical proof of how inherently vague these notions are and demonstrates just how little idea mathematicians have of what these objects they're dealing with are.
>Are there math problems that Mr. Wild can solve that math using the Reals can't?
Wildberger did some cool stuff with rational trigonometry that is typically done by appealing to the real numbers (actually it's typically done in a high school context where the students don't even know what real numbers are) but that probably doesn't satisfy you.
It's hard to understand what you mean by this question as it's so vague. There are some problems you can solve without the reals in some areas of mathematics that are unrelated to analysis, of course, but those are probably not what you're talking about here. It seems like if a problem is not about the reals, then it cannot be an example to your question. However, if it's about the reals, then necessarily a solution to the problem would be using the reals.
If you consider mathematical education a mathematical problem, then yes, not appealing to the reals basic your mathematical ideas on clear, precise concepts can help mathematical education immensely. As probably can his rational trigonometry, which I think contains some results of his own.
>Is there a contradiction in the axioms of the Reals
I suspect, as probably does everyone else, that there are no logical contradictions in the theory. But is the suspicion of consistency the standard by which we accept a foundational framework?

>>12379178
>Then we divide the number of programs that halt by the total number of programs
The number of programs is unending, as is the number of programs that halt. What type of division are you referring to?
You are being very very imprecise with your answer. I would like for you to specifically answer the question of what makes this concept, this "probability" a number. In what sense is it a number? What does it look like?

>>12379178
Also ironically what you're describing seems to be very similar to Emil Borel's notion of an all-answering number, which is one of the main reasons he started to view the "real" numbers are ridiculous and fake.

>>12379209
>The number of programs is unending, as is the number of programs that halt. What type of division are you referring to?
I see you don't like that idea, so let's say we can approximate that number indefinetly by a never ending monte carlo approach. Are you fine with that?

>>12379136
>In general, it does.
No it doesn't. Cauchy examples are examples of sequences that DON'T require an infinite amount of time, therefore sequences don't "in general" require infinite work. You're using the common vernacular definition of "in general" instead of the mathematical definition the same way creationists misuse the word "theory". If you're going to larp as someone who understands math outside of remedial college courses, then start speaking like one.
>A lot of open problems in mathematics could be reformulated as checking whether a certain rational sequence is Cauchy or not.
That's kind of like saying that every problem in engineering can be reformulated into "does it turn on when I press the power button". It's the kind of pathetic oversimplification that only a brainlet could hold.
>This is because the sequences they deal with are always very contrived
What does that even mean? How is "prove or disprove that 1/n is a Cauchy Sequence" a contrived question? This is a serious question. You are using a completely SUBJECTIVE criteria and terminology as though it were an objective one. Knock it off.
>assuming it's meaningful to talk about a general sequence, which is doubtful
It's useful to people who aren't stupid enough to argue that literal undergraduate math homework problems require more than a free afternoon of work at most.

>>12379240
It's not that I don't like your idea, even a given modern mathematician who accepts the orthodoxy regarding the real numbers will tell you that what you just described is vague nonsense. Given that there is an unending number of halting problems and problems in general, you can make the ratio of halting algorithms to general algorithms approach any number between 0 and 1, for example, 1/pi, or 1/(pi*e) + 1/3 * {1 if CH is true and 0 if CH is false} which you probably consider a fine real number, provided you choose an appropriate enumeration of the algorithms.
>so let's say we can approximate that number indefinetly by a never ending monte carlo approach
This is still too vague. How do you approximate it? And how do you guarantee that if another person approximates it like that they will get the same result.
You clearly have no idea what you're talking about lol.

>>12379258
>And how do you guarantee that if another person approximates it like that they will get the same result.
They wont, it is an uncomputable number.

>>12379241
>Cauchy examples are examples of sequences that DON'T require an infinite amount of time
Hahaha where did you get this idea from? As far as I know, there is nothing in the definition that ensures determining a sequence is Cauchy can always be done in a finite amount of work. In fact, it provably can't. If you have an algorithm for determining whether a Cauchy sequence halts, say even an extremely restrictive Cauchy sequence such as one given by an algorithm (which you believe the VAST majority of sequences aren't lol), then you have an algorithm which solves the halting problem, which was proven by Turing to be impossible.
There is no way in general of determining whether a Cauchy sequence converges or not.
>You're using the common vernacular definition of "in general" instead of the mathematical definition
Mathematicians regularly use the word general in the same way I used here. They will say that a general real number is normal (i.e. non normal numbers have measure 0), irrational (the rationals are countable), transcendental (algebraic numbers are countable) and incomputable (computable numbers are countable). None of this is controversial among mathematicians.
In the same way, for a general sequence there is no way to determine whether it's Cauchy in finite time.

>>12379285
>That's kind of like saying that every problem in engineering can be reformulated into "does it turn on when I press the power button". It's the kind of pathetic oversimplification that only a brainlet could hold.
It's a good illustration of just how complicated the checking of Cauchy condition actually is. If some clueless engineer were to assert that he can determine whether any given machine turns on if you press the power button you would be right to point out that a lot of complicated problems in engineering can be reduced to such a problem, illustrating his stupidity.
>What does that even mean? How is "prove or disprove that 1/n is a Cauchy Sequence" a contrived question?
Examples of how it's contrived:
- 1/n is a sequence that converges to 0. Most sequences don't converge to 0.
- 1/n is a computable sequence (a sequence that can be described by an algorithm). Most sequences are not computable..
- 1/n is a decreasing sequence. Most sequences are not decreasing.
It's obvious that 1/n is not at all representative of what a general rational sequence looks like.

>>12379285
>>12379287
Can someone quickly tell me who is one which side here - just so I can follow the thread. You seem to be posting the same sort of pedo avatars

>>12379195
>undefined notions such as a function or a set
Fuck you for either lying or being a lazy retard. Functions are defined in literally every undergraduate textbook on analysis. Christ, it's not even that hard.
And even if I assume you aren't a retarded liar for not knowing what a function is, you didn't answer my question. "These people don't know how something works" is not equivalent to "this something doesn't work". Do you understand that? People didn't understand for centuries why piss left out for several months made for great gunpowder, but they did it anyway because it worked.
>It's hard to understand what you mean by this question
Considering that you don't know what a function is, I absolutely believe that you had a hard time understanding this question. That first sentence about high school trig MIGHT have answered my question, if you bothered to not be VAGUE about it, hypocrite.
>But is the suspicion of consistency the standard by which we accept a foundational framework?
No, but you need a HELL of a lot more than "hur dur what if not lolz" to justify not using the Reals.

>>12379285
>If you have an algorithm for determining whether a Cauchy sequence halts

what

>>12379268
LOL. Your response is very funny.
Here, I also came up with an uncomputable number.
To get the first digit, you go to your neighbor, ask them about their day, encode their answer with some fixed encoding scheme and maybe add 3 if you like. Then put 1 if the resulting number is even and 0 otherwise.
An interesting number you get.
If you ask some other person to perform the same experiment, they might get a different result, but it's actually the same well-defined number. The reason you get different results is that it's an incomputable number.
Even though it's an incomputable number and the only way to get it is to perform an infinitely many experiments which give you random gibberish each time, it's still a perfectly valid number (because it is incomputable) and you are hereby legally required to treat it with respect and dignity just like regular numbers 1,2,3.

>>12379305
My bad. I meant an algorithm for determining whether a given sequence is Cauchy (satisfies the Cauchy criterion).

>>12379293
>pedo avatars
based newfag brainlet

>>12379300
>Functions are defined in literally every undergraduate textbook on analysis
Let me define some things for you. You probably think there is no such thing as a real random number? You think it only makes sense to talk about random variables as measurable functions? Well you're wrong! Here, let me define it for you.
A random number is an irregular number that can be expressed as a sum of 2 primes in at least 4 different ways.
Here's a definition for you. Now you can go into the world with your newly expanded knowledge.
Obviously this is ridiculous. I used an unexplained concept of xhaghuyra in my "definition" which means I haven't defined shit. You have as little idea of what a random number is as you had before.
In the same way, just because you can say "a function is a set that satisfies such and such property" that doesn't grant you the right to say that you've defined what a function is. You never explained what a set is, so your "definition" is as good as the one I gave for the concept of a random number, even though parts of the definition use clear language (such as the property that for one input there is only one output).

>>12379335
>xhaghuyra
Sorry, meant to say "irregular number" here. When I first wrote this I used just a random word but then realized using a word that seems meaningful but actually isn't is better to illustrate my point.

>>12379293
Those are both my replies. I am on the side of Mr. Noseberger.

>>12379285
>There is no way in general of determining whether a Cauchy sequence converges or not.
Dude... it is literally a theorem that convergent sequences are Cauchy, and Cauchy sequences are convergent. Checking whether a Cauchy sequence converges takes literally zero seconds. Thank you for proving that you didn't actually take an analysis class or read a textbook on analysis.
>Mathematicians regularly use the word general in the same way I used here.
And scientists use both definitions of "theory" all the time. The difference is that scientists, unlike you and creationists, don't let it fall into an equivocation fallacy.

>>12379306
>but it's actually the same well-defined number
Well, no, it is not the same neighbour.

I did not say that we will get different numbers because it is uncomputable, I said that the guy will get no number at all because it is uncomputable. In my post I was talking in this ethereal imagination land where we could do something as absurd as a never ending sampling, I thought that was implicit. Then you asked about another guy trying this number, well he can't even evaluate the first one, because the algorithm to evaluate the halting problem cannot exist.

In the real world what you can do is create an encoding scheme for a program, generate all programs in that scheme that are smaller than a certain X, then patiently prove by hand, or use some domain specific programs to solve certain classes that may arise. Then you'll have the Chaitin constant for that X.

>>12379293
>anime style=pedo
Based retard.

>>12379342
Tooker, why are you hiding behind anime girls? You used to like dedekind cuts, remember? You use them in your solution for RH.

>>12379343
I made a typo there. Meant to say whether a given rational sequence converges or not.
Also as for your further points.
>it is literally a theorem that convergent sequences are Cauchy
Yes, if you know that a sequence is convergent, it's very easy to show that it's Cauchy.
>Cauchy sequences are convergent
Cauchy sequences are not convergent in general. For all Cauchy sequences to be convergent you need a property called "completeness". The rationals are not complete. Mathematicians knew this but still wanted Cauchy sequences to converge so bad that they literally started doing fake maths to force them into convergence lol. Now what does a general Cauchy sequence (q_n) converge to? The INFINITE(?), UNCOUNTABLE SET that contains (q_n) and ALL OTHER SEQUENCES(???) (a_n) such that the LIMIT of (q_n - a_n) is 0. If that's not immediately striking as ridiculous, I don't know what is.
>don't let it fall into an equivocation fallacy
How am I committing equivocation fallacy?

>>12379345
> I said that the guy will get no number at all because it is uncomputable
Ok so what IS this number? You can't compute it, you can't even compute the first n digits of this number. So what is it? How is it defined and why does it deserve to be called a number?
Also please explain how your number is so different (if it is) from the following "number": {1 if CH is true and 0 if CH is false}.

>>12379354
I am not Tooker, sorry.

>>12379384
Don't lie to me, Tooker.

>>12379390
I literally see 0 reasons why you would think that I'm Tooker except the fact that my views on mathematics are also unorthodox. Tooker believes in infinities, he even created some, I don't.
Tooker thinks the "real numbers" make sense and he thinks he understands them better than even the contemporary analysts, I don't.
The difference between us is night and day.

>>12379377
>Ok so what IS this number?
Dunno, it is uncomputable.
>You can't compute it,
Yeah
>you can't even compute the first n digits of this number.
Yeah
>So what is it?
An uncomputable number.
>How is it defined and why does it deserve to be called a number?
But I've told you that before. Of course, I've explained it in a fairly non rigorous manner, but it is the spirit of it. You could instead say that the probability is the infinite union of the probability of subsets of this domain of all programs.
>Also please explain how your number is so different (if it is) from the following "number": {1 if CH is true and 0 if CH is false}.
What is CH?

>>12379427
You still didn't explain how you define this number. I pointed out how your first definition was wrong and meaningless, which I assume you understood by now.
So how do you define this "number".
Sure, I get that you can't calculate it, but that's not what I'm asking you about.
I'm asking you what it IS and what right does it have to be called a number.
>What is CH?
It's the continuum hypothesis, i.e. the statement that every proper uncountable subset of the reals is bijective with the reals.
>infinite union of the probability of subsets of this domain of all programs
lol wat

>>12379372
All you’re doing here is proposing an idiosyncratic definition of convergence—that sequences of rational numbers can only converge to rational numbers—then arguing from your own definition. “Completeness” has nothing to do with convergence. It’s a circular fantasy argument.

>>12370345
Praise the Wild Burger!

Next, we retake Constantinople.

>>12370879
Praise God! Death to the infinity Infidels!

>>12370899
Verily the Lord sayeth, you shall burn in the finite fires of HELL!

>>12379477
>that sequences of rational numbers can only converge to rational numbers
If we're restricting our attention to the metric space of rational numbers, this is true. A sequence of rational numbers that doesn't converge to any rational number is not convergent in the rationals.
You can also work in p-adic numbers where convergence is different. The notion of convergence depends on the space that you're considering.
>“Completeness” has nothing to do with convergence
Completeness has quite literally everything to do with convergence. A metric space is complete if and only if every Cauchy sequence in it converges.
The rationals are not a complete metric space space.
The "real numbers" purport to be a complete metric space however. Sadly, none of the constructions of the "real numbers" work, so there's no reason to believe they exist.

>>12371142
What is obvious is you are a GOD CURSED INFINITY LOVING SODOMITE!

>>12374018
Spoken like the seething SODOMITE like you are. You are DAMNED, heretic. DAMNED FOR A LENGTH OF TIME EQUAL TO MOAN.

>>12379505
>>12379514
>>12379493
>>12379491
>>12379482
Stop it. You're cringe and unfunny.

>>12379504
>notion of convergence
Is always about closeness.
>depends
On how we define close. Not on anything else.

It has nothing to do with metric space.
It has nothing to do with completeness.

>>12379532
In incomplete spaces there are Cauchy sequences that don't converge.

>>12379516
Listen SODOMITE, there is nothing uncringe or funny about facing FINITE DAMNATION in the DISCRETE fires of HELL.

CONFESS HERETIC! Confess your many mathematical sins and your worthless soul may yet be worth saving.

>>12379541
Not true. They may not converge to an element of the space, but they always get arbitrarily closer and closer to each other—i.e., they converge—by definition.

Completeness has nothing to do with convergence. There isn’t even any overlap in what those properties try to describe.

Now we have established a firm foundation of the ONE TRUE FAITH of the DISCRETE and FINITE Universe, we must look to expunge all heresy from the world.

I think we should start with book burning. Yes. Any books which contains any reference to infinity will be burnt, along with their authors.

>>12379555
What do you understand the statement "A metric space X is complete" to mean?
So in your mind there is no difference between a sequence that is cauchy and a convergent sequence? You think there are no sequences that are Cauchy but do not converge?

>>12379551
>Confess your many mathematical sins
I confess to not knowing what a "set" is. But in my defense, I'm pretty sure nobody else knows what it means either.

>>12379443
>I'm asking you what it IS and what right does it have to be called a number.
As I said before, look >>12379178
>It's the continuum hypothesis
Oh, ok. Well, then I don't know, because you have to prove that the solution to CH is uncomputable.

>>12379573
Sets are like... burritos.

>>12379568
>metric space X is complete" to mean?
That all Cauchy sequences converge in X. What do you imagine it means?
>You think there are no sequences that are Cauchy but do not converge?
It’s literally part of the definition. I’d be fascinated to see what you’ve imagined as a counterexample.

>>12379178
>>12379579
>Then we divide the number of programs that halt by the total number of programs
I already explained to you how that doesn't work.
Also, your definition depends on asking God whether a program halts or not. That's not possible to do in general. If it were, you would have an algorithmic solution to the halting problem, reaching a contradiction in the fabric of reality. If your assumption leads to a contradiction, it means your assumption is wrong, thus your "definition" cannot rely on asking God whether a given program will halt.

>>12379568
>>12379555
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

>>12379586
Please prove to me that in a metric space X, if a sequence (x_n) is Cauchy then it is convergent. I'll wait.

Dedekind cuts or Cauchy sequences.
In case you have to choose between those two, which is better?

>>12379592
I’m already first in line, waiting for your counterexample of a Cauchy sequence whose elements don’t get arbitrarily close to each other.

>>12379593
They're both equally bad. Neither of them work as a "construction".

>>12379593
Cauchy sequences are more kino, Dedekind cuts are more rigorous. If you care at all about the construction of the Reals you are a faggot and will never contribute to mathematics. If you care about the Real number system or go on some crusade to expunge continuous math from science and engineering you are a brainlet or a schizophrenic. Reality is local, continuous, has probabilistic representations at small scales, and is most importantly physical ie. mathematical models do not do justice to the Real thing. Any good physicist and for that matter any good scientist would agree with every point. You have some people like Penrose who would assert that math is more fundamental but this is not the prevailing view.

>>12379606
>Cauchy sequence whose elements don’t get arbitrarily close to each other
That's literally what being Cauchy means. It's the definition of the word Cauchy. However, a sequence being convergent means that there exist a point to which it converges, i.e. to which it gets eventually arbitrarily close to.

>>12379613
>there exist a point to which it converges
No, that’s not what convergent means. That’s your own idiosyncratic hybrid definition, into which you conflate the unrelated concept of completeness.

>>12379608
>Dedekind cuts are more rigorous
How? They're so obviously equally bad.
It's also harder to prove laws of arithmetic using Dedekind cuts.
>If you care at all about the construction of the Reals you are a faggot and will never contribute to mathematics
Wildberger published respectable mathematics.
According to you Borel, Cantor, Kronecker, Brouwer, Weyl, Dedekind, Cauchy were all faggots who never contributed to mathematics.
>expunge continuous math from science and engineering
The issue at hand is not about continuous math but rather about how to model the continuum and whether the current model of "real numbers" is any good (it's not). There's nothing inherently wrong about continuous math. Also the math used in science and engineering is completely finitistic and computable.
tl;dr you're a faggot retard who needs to go back to his containment website

>>12379628
LOL. What's the difference, according to yours truly, between a convergent sequence and a Cauchy sequence. You said there is a proof that these two notions are equivalent, which means they must be formulated in a different way, otherwise proof is unnecessary. So what is the meaning of these terms?

>>12379588
>I already explained to you how that doesn't work.
You disliking doesn't mean it is not possible under the mathematics that the academic world has chosen.
>Also, your definition depends on asking God whether a program halts or not. That's not possible to do in general.
Yes, because it is uncomputable.
>If your assumption leads to a contradiction, it means your assumption is wrong,
My assumption is that all programs fall under one of the two options: halts or does not halt.
> thus your "definition" cannot rely on asking God whether a given program will halt.
TREE(3) is finite and integer. TREE(4) is finite and integer. Is TREE(3)*TREE(4) odd or even? Well, it is one of the two for sure, but we cannot do that multiplication due to our physical limitations. That works the same for a turing machine and a program. Does the program halts or not, Turing Machine?
-(Turing Machine) geez anon, I don't have a fucking clue
See? We are the same.

>>12379633
>According to you Borel, Cantor, Kronecker, Brouwer, Weyl, Dedekind, Cauchy were all faggots who never contributed to mathematics.
You didn't CONSTRUCT the reals you fucking nigger, you learned about them in a freshman analysis class, unless of course you didn't because you aren't even conversant in the most basic definitions taught in those classes.
>Also the math used in science and engineering is completely finitistic and computable.
No it isn't retard.
>containment website
Are you sure you shouldn't be sucking down the contents of a mossberg right now faggot? You seem like you really don't like the way other people have been doing math and science for much longer than you've been alive. Maybe if you killed yourself you wouldn't have to make retarded bait to cope with the fact you can't contribute to anything.

>>12379642
>according to yours truly,
Are you asking me what I think the difference is according to you? How should I know what you think?
>You said there is a proof
Huh? I never said anything about a proof. Are you okay?
All I’ve said is that convergence is when things get arbitrarily close, and that you can’t actually describe a “Cauchy sequence” that doesn’t converge, because convergence is in the definition itself. I’m still waiting for your example.

>>12379648
>You disliking doesn't mean it is not possible under the mathematics that the academic world has chosen.
I explained to you how choosing different enumerations of the programs and taking the limit gives you different answers.
You haven't even said that you're taking the limit of anything. You just said you're dividing one unending quantity by another unending quantity, which makes 0 sense.
Then you said something about Monte carlo simulation, which obviously doesn't work either as a definition of your "number".
>My assumption is that all programs fall under one of the two options: halts or does not halt
That's fine and logical.
>TREE(3) is finite and integer. TREE(4) is finite and integer.
Not sure if these are true, I haven't looked into this stuff, but I'll take your word for it.
>Well, it is one of the two for sure, but we cannot do that multiplication due to our physical limitations.
Sure, that's reasonable.
So again, tell me what IS this number, this time be precise. And explain what right it has to be called a number.

>>12379608
meh, you can't at the same time assert that some people are cringe and shizo, while also saying that no serious scientist would would disagree with (insert my option here)

>>12379656
Seething.
>>12379671
> I never said anything about a proof.
> it is literally a theorem that convergent sequences are Cauchy, and Cauchy sequences are convergent.
A theorem with no proof? Curious.

>>12379674
The overwhelming consensus view among theoretical physicists agrees with every point. Reality is continuous, quantum phenomena are probabilistic, the world is local. I never said anyone was cringe you little faggot I said that you're either retarded or a schizophrenic if you disagree with this. A very small number of physicists disagree with these premises, they are in the minority at the moment, for instance Roger Penrose is one of them.

>>12379633
>the math used in science and engineering is completely finitistic and computable.
that's a stretch, at least for physics models

>>12379672
>>12379672
>So again, tell me what IS this number, this time be precise.
A transcendental, real, uncomputable number.
>And explain what right it has to be called a number.
Watch your numberism, bigot.

>>12379686
He doesn't know anything about physics or math, he doesn't even know lower division analysis. He doesn't even know what a complete metric space is or how to prove that a sequence is convergent. It's just a bait thread by a brainlet tourist.

>>12379684
>you little faggot
what's your problem, work on your anger issues
this is /sci/

>>12379687
>A transcendental, real, uncomputable number.
Is it the only one such number? If not, how is this a definition?

>>12379689
Say that to my face and not online pussy

>>12379688
> it is literally a theorem that convergent sequences are Cauchy, and Cauchy sequences are convergent.
Tell me more about this theorem. How is it proved?

>>12379682
> it is literally a theorem that convergent sequences are Cauchy, and Cauchy sequences are convergent.
Instead of making up quotes and ascribing them to me, perhaps you can simply provide the counterexample you promised, which I’ve been patiently waiting for now for the last half hour or so.

>>12379704
It was a continuous stream of conversation so I assumed it was you. You should preface your posts by "not that anon" if you want me to make a distinction.
>provide the counterexample you promised
Sure. The rational numbers in the rationals is a metric space with the following metric
d(x,y)=|arctan(x)- arctan(y)|.
The sequence x_n = n is a Cauchy sequence in this metric space and yet is not convergent: there is no rational number to which it converges.

>>12379715
>yet is not convergent: there is no rational number to which it converges
You are describing incompleteness, not divergence.

The Burger of the Wilderness is on the right track by questioning the foundations, but he does not go far enough.

I question the entire foundation of cardinal numbers.

We extract a number from the Universe without acknowledging that by this very process we have committed the act of subtraction. By counting "1" the Universe now contains 1 less of what it had.

Also there is no place for negative numbers. Fuck negative numbers.

>>12379736
Yes, the space is incomplete because the sequence I gave is Cauchy yet doesnt converge.

>>12379693
One day SODOMITE you will stand ( more like crawl ) before GOD ALMIGHTY and be judged not only for your anger issues but also for your HERESY!

>>12379749
No, the space is incomplete because the sequence doesn’t converge to an element in the space.

>>12379755
Yes, it's a Cauchy and it doesnt converge.

>>12379764
Of course it converges. Its elements get arbitrarily close. The fact that the elements in the sequence don’t get arbitrarily close to an element in the space means that the space isn’t complete, not that the sequence doesn’t converge.

>>12379785
What's the difference between saying "A sequence converges" and "A sequence is Cauchy". How are these two notions different?

>>12379796
They aren’t. All Cauchy does is describe convergence in terms that doesn’t require a limit belonging to the same set of values as all the elements of the sequence.

>>12379898
What does the sequence x_n in
>>12379715
converge to?

>>12379925
What are the elements of the sequence you’re asking about?

>>12379947
The n'th term of the sequence is the rational number n.

>>12379743
woot

Edward Nelson thought PA is formally inconsistent, because the complexity taken to compute certain iterated successors of 0 exceeds the number it denotes (or something along those lines),
but that's not really a "take away from the universe" argument like yours

>>12379950
The 1st term of the sequence is the rational number 1? What sequence are you actually trying to describe with your d() |tan^-1| notation?

>>12379964
>The 1st term of the sequence is the rational number 1?
Yes.
>What sequence are you actually trying to describe with your d() |tan^-1| notation?
d is the metric I defined on the rationals.
It's not part of the sequence. The distance between two rational numbers x and y are defined with this metric to be
d(x,y) = |arctan(x)- arctan(y)|.

>>12379978
Your still not defining a sequence. What are the first 3 elements of the sequence?

>>12379990
The first 3 elements are 1,2, and 3.

>>12379978
You’re still not defining a sequence. What are the first 3 elements of the sequence?

>>12380001
Sorry, just correcting the typo.
>The first 3 elements are 1,2, and 3.
What are they, though.

>>12380006
They're rational numbers. Do you not know what a rational number is?

>>12379978
>>12380002
What’s your Cauchy sequence? Are you trying to describe the sequence of values that gets arbitrarily close to 0 for arbitrarily large values of N, and whose sum over N gets arbitrarily close to π/4? Am I reading your mind correctly on this?

>>12380037
>>12380045

>>12380045
The sequence is literally x_n=n.
It inputs a natural number n and returns the natural number n.
The elements of the sequence are natural numbers embedded in the rational numbers. The sequence is Cauchy but does not converge.
As another example, you can take the sequence
x_n=2^n which is also Cauchy wrt my given metric but doesnt converge.

>>12380061
>The elements of the sequence are natural numbers embedded in the rational numbers
This is meaningless.
>The sequence is Cauchy
Nothing you’ve described is Cauchy.

>>12380073
Lol ok I realize im talking to an idiot. Have a good day sir.

>>12380073
Anon, this is not the guy you are arguing with, but what he is saying is absolutely correct and you are being a dumb. His wildburger ideas are retarded and I can understand if you aren't paying too close attention to his words on that basis, but what he's saying about Cauchy sequences here is entirely right. Anon has defined a perfectly meaningful sequence in a perfectly meaningful metric space, and that sequence is indeed Cauchy (and not convergent). Try reading it again, calmly, without the (understandable) basic assumption that whatever he is saying must be nonsensical.

>>12380076
Lol, all you’re doing is inventing your own idiosyncratic language and then complaining that your solipsistic jargon doesn’t properly describe whatever math concept you imagine you’re describing.

>>12380082
Since you seem to have had more success reading Anon’s mind than I, can you explain how the elements of whatever sequence you see are getting arbitrarily close but aren’t “converging.”

>>12374916
>which doesn't work because the notion of a "sequence" is always left undefined.
>>sequences are undefined
damn, imagine being such a pussy you cant even make a function from N to a set

>>12380111
He is using the metric space consisting of the rational numbers with the distance metric d(a,b) = |arctan(a) - arctan(b)|. He is defining the sequence f_n = n, i.e. [0, 1, 2, 3, 4, ... ]. That is, the natural numbers in the usual order.

The expression arctan(x) gets increasingly close to pi/2 as x gets closer to infinity. Or to be more precise about it, for the regular real numbers, the limit of arctan(x) as x tends to infinity is pi/2. That means that for two very large natural numbers x and y, arctan(x) and arctan(y) are both very close to pi/2, which means that |arctan(x) - arctan(y)| is close to zero. For example, |arctan(10000) - arctan(1000)| is about 0.0009, and |arctan(1000000) - arctan(100000)| is about 9*10^-6.

In particular, for all epsilon > 0, there is an N, such that for all natural a,b>N, |arctan(a) - arctan(b)| = d(a, b) = d(f_a, f_b) < epsilon. This means that f is a Cauchy sequence *in the metric space defined by the metric d*.

But there is no rational number Q that is the limit of the sequence f under this metric. This means that f does not converge to any point in the metric space defined here, AKA does not converge, period.

If you squint, you could say that f has a limit at infinity, and f converges to this point outside the metric space. But that doesn't actually mean anything, because the distance metric d() isn't even defined for infinity, it is defined only for rational numbers. For this reason, the notion of "this sequence converges to something, but that something is not in the metric space" is bunk. Convergence is meaningful only relative to the space it's in, because that's the only place where the distance metric is defined. For that reason, the standard terminology here is that "f does not converge", full stop.

The sequence f does do something kinda like converging, and that something is "being a Cauchy sequence". Which is why "Cauchy sequence" and "convergent sequence" are two different things.

>>12380167
So the Cauchy sequence is |tan^-1(n+1) - tan^-1(n+1)|? That certainly converges. As you said, it gets arbitrarily close to 1/4 the circumference of the unit circle, which we shorthand using the symbol π/2.

>>12380191
* |tan^-1(n+1) - tan^-1(n)|
forgot to edit the cut & paste

>>12380191
No, the Cauchy sequence is [0, 1, 2, 3, 4, ... ]. That is the sequence we're working with here. d is the distance metric, that defines when two numbers are and aren't close.

This is a notion from metric space topology. Have you studied metric space topology before? If not, I'm afraid anon's argument will be over your head.

>>12380209
Of course, the “argument” is to cut the space at the limit, and define it as out of bounds (using < as a glass ceiling) rather than in bounds but forever in between the elements of the space. That’s not really an argument, and has nothing to do with the fact that the sequence always converges to the same value, no matter how you frame it.

>>12380249
>Of course, the “argument” is to cut the space at the limit, and define it as out of bounds
Yes and no. Yes, that is how we defined this particular space, but that is only obvious from the viewpoint of a person that already knows a broader space that does include that limit. The situation isn't quite so artificial in more practical cases, where you start with a space that doesn't seem to have all limits, and where you must do real work to create an extension of that space to artificially create those limits.

>That’s not really an argument
Sure it is. You can't really study the structure of the rational numbers without it.

>and has nothing to do with the fact that the sequence always converges to the same value, no matter how you frame it.
Indeed, it has nothing to do with that. If a limit of a sequence exists, that limit is unique; that part has not changed. But that does not mean every Cauchy sequence has a limit.

>>12380260
You can study the rationals equally well in terms of limits that are forever in between. You don’t need an artificial glass ceiling at infinity to do that.
>that does not mean every Cauchy sequence has a limit
Can you describe one that doesn’t? (Given that I disagree with the last example. Or are all of the examples you have in mind parlor tricks like that?)

>>12380285
>You can study the rationals equally well in terms of limits that are forever in between.
I'm not sure what this means.

>Given that I disagree with the last example. Or are all of the examples you have in mind parlor tricks like that?
I suspect you'll judge that they are. You see, the real numbers with the usual distance metric are complete, which means that every Cauchy sequence in that space has a limit point (i.e. it converges). Which means that every Cauchy sequence in the RATIONALS I come up with, will always have a limit in the REALS, because the reals are a complete extension of the rationals. If you are going to reject every such case as misleading tricks, then I fear you'll need to study some topology proper before any examples will make sense to you.

>>12380296
>I'm not sure what this means.
You described it in the next paragraph. Rational Cauchy sequences whose limits are forever in between closer and closer rational values but aren’t rational values themselves. Completeness isn’t the same thing as convergence. The concepts don’t even overlap. You and the other Anon are conflating two distinct ideas. Convergence is simply getting closer and closer.

>>12380316
>Rational Cauchy sequences whose limits are forever in between closer and closer rational values but aren’t rational values themselves.
Okay, I think I understand what you are getting at. Consider, then, the example of f_n = (the largest n-decimal rational number x such that x*x <= 2). In the real numbers, this is a Cauchy sequence with a limit the square root of 2; in the rational numbers, this is a Cauchy sequence without a limit, i.e. a sequence that does not converge.

This is the sort of example you had in mind, right?

>You and the other Anon are conflating two distinct ideas.
Are you sure you understand what completeness means, anon?

>Completeness isn’t the same thing as convergence.
Very true. Convergence is a property of sequences; completeness is a property of metric spaces. A metric space is complete if and only if every Cauchy sequence converges. The reals are complete, the rationals are not.

Math major chiming in. I've taken analysis. I skimmed the video. The video is just wrong. Every individual claim it makes can be easily refuted. If anyone doesn't believe me feel free to reply and I'll explain why a specific aspect is incorrect.

>>12379305
Why isnt there a currency we measure in complex numbers? 6 dollars, 42 cents, and 10.5 imaginary units.

>>12380343
Wildburger is a well-known math crank, anon. Everyone knows his stuff is nonsensical. These threads are just people pretending to take him seriously to show how edgy they are.

>>12380334
A metric space is complete iff every Cauchy converges to an element in that space. All Cauchy sequences converge—they get arbitrarily close by definition. If a space is incomplete, it doesn’t mean that some Cauchy sequences don’t converge, it means that the values they do converge to aren’t necessarily in the space.

>>12380379
I already told you that is not what convergence means, anon. Repeating it doesn't make it true.

>>12380385
You just accused me of arguing to my own anonymous authority by arguing to your own anonymous authority. That also has no truth value, and is self-contradictory to boot.

>>12380401
I accused you of nothing, and appealed to no authority. I'm just stating the fact that you are mistaken. It's true that there's not much that can be argued about this, and indeed if you are not going to listen then this discussion is going nowhere.

>>12379608
it's retarded to think maths are about reality in the first place

>>12380348
>>12380343
yep undergrads who think they know maths are retards