>yfw modern mathematics is inconsistent but mathematicians are too lazy to tear everything down and build it up back up again so they just roll with it and pretend everything is alright
>>7720904
please explain how modern mathematics is inconsistent
>>7720905
>1=1+1
>>7720907
>Implying that an object composed of an uncountablly infinite number of points is "1"
>Not realizing that the Banach-Tarski "paradox" requires the axiom of choice specifically
>trying to simplify a fascinating mathematical theorem into a retarded arithmetic statement
>>7720904
>>7720907
>yfw this fool doesn't understand the paradox/theory
>yfw he claims it's inconsistent because of latter
>just viper yourself my man
>>7720914
>Implying that an object composed of an uncountablly infinite number of points is "1"
.000....0001 + .000....0001 + .000...0001 +.....+.000...0001+... = 1
nice try though doofus
>>7720923
i have a degree in anime you fuck
>>7720905
2000 years ago.
Please explain whats wrong with using roman numerals?
Infinity? What an absurd concept!
You want to introduce something called "zero"?
>>7720927
>.00000...0001
That's an impossible number.
If the zeros go on forever, there's no room for that trailing 1
There's either a finite number of zeros before it, or there aren't. And if there aren't, that terminating 1 cannot exist.
>>7721075
You are autistic anon you dont understand what op is trying to say and yes modern math is fucking wrong you cant just duplicate an object this sounds like a fucking joke
>>7721089
I'm not saying that OP is correct.
The problem is that the set theory in the OP assumes the axiom of choice. Which is wrong.
By the way, the axiom of choice has been frowned upon by modern mathematicians. There isn't currently a rigorous proof to show that makes a contradiction, but everyone pretty much agrees it's a hack of an axiom that causes all sorts of problems.
>>7721089
Math doesn't have to model the real world, though it's nice when it does.
>>7721075
I am happy to inform you that you are fucking wrong and that it is easy to construct a number system which consents such numbers.
>>7721127
Not an internally consistent number system, no.
Infinity exists as a concept where variables approach a boundless value. Infinity cannot exist on the number line either. There is literally no room to put it anywhere. It's like placing a stone at the bottom of a bottomless well. No such place exists for it.
>>7721096
lol what.
>frowned upon by modern mathematicians
There's a few heterodox guys out there (inb4 WILDBURGER floods this thread), but most mathematicians accept choice as a pretty reasonable axiom.
>>7721170
I'm not going to TeX this out so just bare with me.
Consider the set of functions from N×N to Z/nZ under componentwise addition and multiplication. This naturally forms a Z-module whose elements may be written (whimsically) in the same form as 0.01010001...01010001.
>>7721182
For any finite N, you're conveniently leaving out.
>>7720904
If you're implying that one of the faults is Banch-Tarsky, then you need to understand what infinite complexity is. For all intents and purposes, the "problems" with mathematics occur in hypotheticals and impossibilities.
>>7721170
Go ahead and read up on the real projective line, my man.
>>7721193
>Go ahead and read up on the real projective line, my man.
I know all about them. It's used for solving problems relating to perspective projection. Notice how they points at infinity aren't actually points that exist on the 3-D plane, but points are points on the 2D plane as the limit approaches infinity on the third axis. That is why they require hyperplane, and why such points are called ideal points. These are just shorthand terms.
>itt: no one has taken any math beyond intro to analysis
>>7721187
No, N was meant to indicate the natural numbers.
>>7720904
serious mathematics hasn't given a shit about axioms since the 40's
>>7721216
I was just talking about modules anon.
>>7721218
Infinity is not a natural number.
In fact, it's not a number at all.
>>7721227
Do you know what sets are? Am I being trolled right now?
>>7721225
Anyone can write some drivel about a term you learned by reading the Wikipedia page for five minutes. You never really bothered to explain how your "construction" leads to a rigorous conceptualization of infinitesimals.
>>7721237
That's cute. I never said anything about infinitesimals, but if you're interested there is a well-founded formulation of calculus in terms of infinitesimals. Look up nonstandard analysis.
>>7721182
>set of functions from NxN to Z/nZ
What functions, specifically?
>under componentwise addition and multiplication
What do you mean by this?
>This naturally forms a Z-module whose elements may be written (whimsically) in the same form as 0.01010001...01010001.
"Define the map from Mochuzuki's log-theta lattice to the category of adeles over the sporadic groups... this whimsically proves the Riemann hypothesis."
>>7721241
So if you're not talking about infinitesimals, what exactly are you trying to explain? You do realize there's no such thing as 0.00...001 right?
>>7721240
It varies from place to place. I never saw them until grad school.
>>7721096
You have no idea what you are talking about, pretty much all of modern math uses it.
>>7721243
I don't understand why they refuse to understand this concept.
Either the zeros go on forever (and by extention there is no terminating ...01) or there does does exist a terminating ...01 which implies that the number of zeros is finite. Either way, an infinitesimal value isn't actually a number. It's used as a limit as some value approaches zero.
It's like implying that 0.9999... =/= 1 because there exists some 0.000...1
Which, of course, we can all agree, there doesn't.
>>7721242
The set of all such functions.
Write each function f as a tuple.
(f(0,0),f(0,1),..,f(1,0),f(1,1),...)
Reorder the tuple to have it come to an "end".
(f(0,0),f(0,1),...,f(1,1),f(1,0))
Choose your favorite ordering for the terms in the middle, it doesn't matter because this is just a notational trick. Hence the "whimsy".
Fuck off Norman.
>>7721257
>Reorder the tuple to have it come to an "end".
You will never be able to finish re-ordering an infinite number of them. Again, it will only ever work for a discrete, finite, N.
>>7721265
Come back when understand what functions and sets are. What do you think N×N is if N is an integer instead of the set of all naturals? What is a function from this N×N to another set?
Also, why do you think I should have a problem putting a total order on any goddamn set that I want?
>>7721273
I assume that you understand that when I say "some finite N", what's meant is a finite set, or in other words, finite cardinality.
Consider this: re-order the set such that the last element comes first, and assume that every element is unique.
You'll get: (f(inf,inf), f(inf,inf), f(inf,inf), ....)
Also, you didn't address this: >>7721254
>>7720904
Finding out math/formal logic is incomplete didn't stop anyone why should this?
No matter how many paradoxes and situations where its obvious we don't know wtf is going on come about people will just continue doing shit and acting like we do know. The alternative is throwing our hands up in the air and saying "idk I quit!". We might not really know what we're doing with math and science but it has made us some neat stuff so its not like its useless even if we are "wrong" about everything.
>>7721281
That order is not well-defined, because the original order had no maximum element. Also you f(inf) stuff is totally meaningless.
>>7721287
>original order had no maximum element
Which is exactly why you cannot re-order it.
There cannot be some terminating element when you declare that the cardinality not finite to begin with.
>>7721283
There is no problem. You can duplicate spheres made of rational points even more easily without AoC. No one is saying the rationals have a problem.
>>7721290
https://en.m.wikipedia.org/wiki/Sharkovskii%27s_theorem
We reorder infinite sets in this fashion all the time. It's fine.
>>7721309
I am not >>7720927
>>7721254
I don't know what 0.9999... or 0.000...1 mean in this context.
I'll give a better version of the construction from before. Let $S=\mathbb{N}\times\{0,1\}$. Let $T$ be the set of all functions from $S$ to $\Z/2\Z$ (the integers modulo 2). For a given function $f\in T$, denote $f$ by $(f(1,0),f(2,0),f(3,0),\dots,f(3,1),f(2,1),f(1,1))$. There is eactly one such tuple for each function in $T$, and two tuples can be added by adding their corresponding components in $\Z/2\Z$.
>>7721344
Ah I shouldn't have even tried to TeX here, I fucked it up and slipped back into using commands from my personal LaTeX package immediately.
>>7721139
this is his channel is someone is interested:
https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
nice meme btw
This is the dumbest thread I have ever read.
>>7721443
Damn right you are! This thread blew my mind.
![GU8wd[1].jpg](https://i.imgur.com/dvij7Qb.jpg "GU8wd[1].jpg")
Explain this mathfags. Excatly what will be the limiting figure and what is it's circumference?
>>7721841
Circles obviously don't exist.
>>7721841
Length of a limit =/= limit of length in this case
>>7721089
Have you even looked at the paper let alone understood what the theorem says? I thought so, now fuck off
>>7721841
You don't get a circle with that construction. The step from 4 to 5 is a total hack.
>>7721100
THIS^999999999
>>7721841
If you can prove that thing actually converges, then the final figure is really fucking weird, it's not a circle.
>>7721100
If it's not modelling the real world then what is it modelling? Some made up bullshit? If I was dictator I'd have anyone who taught any maths that isn't applied gassed.
>>7722423
Math isn't science. It doesn't "model" anything. It gives you axiomatic conclusions from abstract structures that satisfy certain abstract properties. They aren't real, they're ideal and they don't exist, because anything in any possible universe that has those properties will fulfill the same conclusions.
So whenever you find something in our world that can be modeled with a structure that has familiar properties, you can apply these conclusions and instantly have a much, much more powerful model.
Of course that's the "applied" interpretation of math. Pure math in itself doesn't have to care that what it studies is applicable - it might one day, but we don't care. The important thing in pure math is to gain more tools and theory to study certain structures just for the sake of it. It's like a game for me at least. It's challenging, it's fun. It also enhances what applied mathematicians can do.
Why would you gas pure mathematicians? Would you gas artists? Would you gas musicians? Writers? People don't really care that you don't like a field, your opinion has much less impact than you think.
>>7722423
Any math that is taught is applied, the only thing that most of it is applied to other math.
>>7720904
>Set theory is math
>Math is only set theory
>Incompleteness means inconsistency
pls engineer remove yourself
>>7722443
this
>>7721096
>axiom of choice
>frowned upon
-2/10
>>7722443
Natural numbers, integers, rational numbers, real numbers and complex numbers model quantities.
Plane geometry models the planet on small scales.
Logic models thought and how the universe works.
Graph theory can be used to model MANY things including maps and networks.
Differential equations model numerous things.
Science is empirical, I think you mean. Math is empirical - guessing and checking is part of mathematical research before giving a proof of some fact. Problem solving is empirical. Formulating a new system of logic is also often empirical before it is formalized.
>>7721841
this work with regular polygons. If the polygon is inscribed and the circle is of diameter 1 then you solve
[math]\lim_{n \to \infty }n\tan{\frac{\pi}{n}}[/math]
and if the circle is inscribed you solve
[math]\lim_{n \to \infty }n\sin{\frac{\pi}{n}}[/math]
both get you the same value. can you guess it?
>show someone this gif
>when they call you a troll show them it's a proven theorem
>slap them in the face and call them an idiot for ever doubting you
>>7720907
>((log(e^(iπ))+(2iπ))/(ilog(e))((log(e^(5 (403-3i 10^(-e))))+(((2^2^2)-1)i/10^e))/(log(2(e^(-(i^(-e) pi)/((2^2^2)210)))((e^((3 i^(-e)π)/5))/2))*5i^e)))
nigger please get reals
>>7725398
i think you're mixing the construction of the image that does not converges into a circle and a regular polygon of n sides that does. The image is wrong.
>>7725366
Since I know how this works, I can see the diagonal part extending as it moves up and to the right. This was such a mindfuck until you realize exactly what was going on.
