/sci/ - Science & Math

>>12511294
i-inf-f-finity...

>>12511294
Obviously is impossible to have a finite realization of a infinitary concept. The problem is not the infinitary concept, is wanting the finite realization, just like wanting an eternal being to give up its inmortality and have a human time scale existence.
>You can't have a pie and eat it too.

>>12511294

Real numbers are ultimate states constructed by using a supertask to follow a Cauchy Sequence.

>>12511326
based and wildberger pilled

>>12511294
Some faggot Cauchy accidently made them all when he literally made up his own way to check whether a sequence converges.

>>12511294
Stop trying to think about things in terms of "infinite" and start thinking of it as "arbitrary".
When someone says there are "infinite" real numbers between 0 and 1, what that means is that for any two distinct numbers you give me, I will be able to name another number that is between them. And the two numbers you pick can be "arbitrary".

>>12511294
Infinity is actually retarded, I don't know how people get spoon fed this crap.

>>12511294
We do this every day

>>12511489
what do you mean ""name another number""? you can't just make numbers up faggot

>>12511294
As far as I’m concerned you could take the real numbers as an axiom and it wouldn’t make a single fucking difference. Obsessing over the “most rigorous” construction of the reals is one of the most Jewish things I have ever seen on this board, and there is a lot of Jewish shit here. So basically, you are a faggot.

They're forms of quantities

>>12511533
take ur meds infinitist schizo

Explain to me what a natural number is without using a circular definition (i.e. strokes on a whiteboard).

>>12511551
God is infinite, so a belief in the existence of infinities, even if only abstract ones, is completely valid.

>>12511559
natural numbers are a divine axiom, this is a well known fact

>>12511605
t. schizo

>>12511549
No they're not you Platonist fuck. Numbers are material, empirical things in nature that we evolved the capacity to see and use.

>>12511646
>numbers are material, empirical things
>evolved
Retard.

>>12511294
Real numbers are countable abstractions based on the axiom that there can be more than 1 object and that enumeration of said objects builds on a series of whole 1 objects.

>>12511650
with you on this one.
Numbers will work the same in all possible sane universes.

>>12511294
Schizo =/= thing I'm too stupid to understand.

>>12511294
All Complex Numbers with b=0.

>>12511652
>Real numbers are countable

>>12511489
That's true of ANY two distinct numbers, which is only possible if there are an infinite number of numbers in there.

>>12511689
1, 2, 3, 4, 5, 6, 7, ad infinitum. Seems pretty countable to me.

>>12511696
You missed some retard

>>12511696
>Confusing the Naturals and the Reals
Why do I come here? It's filled with literal retards.

>>12511654
>sane
sanity you mean
human reason you mean
evolved brain???????

huh huh huh?

>>12511696
i have brain damage from reading ur post

>>12511294
Each real number is defined by a Cauchy sequence, or equivalently by a dedekind cut

>>12511793
holy fuck do people like this actually exist ?
MEDS NOW

>>12511793
HAHHAHAHAHAHAHAAHAHAHAHAHAAHAH

>>12511802
What exactly is the problem with what I said?

>>12511802
>>12511809
Okay, the Cauchy sequence defined by
[eqn]
a_0 = 1
a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}
[/eqn]
defines the real number [math]\sqrt{2}[/math] since it is the limit of the sequence.
I'm not asking you to calculate the [math]a_\infty[/math] term of the sequence, in fact I'm not asking you to calculate any terms in the sequence.
I have already defined the sequence in a finite number of steps, and the sequence itself defines the real number [math]\sqrt{2}[/math], therefore I have defined the real number [math]\sqrt{2}[/math] in a finite number of steps.
In a more mundane way, I can define the real number 2 from the Cauchy sequence of rational numbers 2, 2, 2... (i.e. [math]b_n = 2 , \, \forall n \in \mathbf{N}[/math]).
Again, I have defined the sequence which itself defines the real number without completing a supertask, or imagining I have completed a supertask.

sqrt(2)

this comes from finite natural numbers, OP.
what's its value?

>>12511851
>>12511802
>>12511809
first equation was meant to be
[eqn]
a_0 = 1 \\ a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}
[/eqn]

>>12511294
Not imaginary and not polynomials

>>12511851
>>12511855
Okay, but what if I have another sequence that also "converges" to "sqrt(2)"?
As a concrete example, define:
a_0 = 1
a_n = a_{n - 1} + k_n * 10^-n
Where k_n is the nth decimal digit. k_n can be found by just trying a_{n - 1} + {0,...,9} * 10^-n and determining the largest value for k_n that doesn't allow the expression's square to exceed 2. This is also a sequence and should intuitively also converge to sqrt(2) right?
Looks like you're going to need an equivalence relation. The standard one between two sequences is:
e(s_n, s_n') = [lim(n -> inf) (s_n - s_n') = 0]
Now how are you going to check whether your sequence and my sequence, or in general, any arbitrary sequence converges to the same value. Well, you're going to have to determine:
For all e > 0, there exists m, n >= N s.t. |s_m - s_n| < e
for the difference between the sequence you gave and any other sequence. Well, since sequences are modeled as choice functions, you have, in general, an infinite amount of work ahead of you (you're not always going to be able to express N in terms of e). You can't escape the intrinsically infinitary nature of the construction of the real numbers.

>>12512227
So what?

>>12512244
So you're still doing an infinite amount of work.

>>12512249
And...?

>>12512253
brainlet schizo detected, meds now

>>12511363
Exactly

Real number definitions are very vague and I understand Norman W.'s personal crusade against them.

A) Bounded-from-above set has a least upper bound.

D) Real number is the rational numbers less than it.

Very vague and inprecise.

>>12512260
based

>>12512255
>I have no argument.

Did you find a contradiction in ZF yet?

>>12512270
your """argument""" is literally just that you can do an infinite amount of work because you assume you can through the axiom of infinity.
that is not an argument; at best it is wishful thinking, and less charitably, it is delusional.

>>12512260
>dedekind cuts
let me know when you are done checking an infinite number of conditions to determine the membership of all rational numbers for the set defining, pi, e, etc.

>>12512278
Don't blame me, I'm just a messenger.

>>12512278
I didn't make a claim. You are the one making claims, so you had better back them up. You fail to do so.

>>12512285
which post is this in the context of?

>>12512278
>let me know when you are done checking an infinite number of conditions to determine the membership of all rational numbers for the set defining, pi, e, etc.
You don't need to, dummy. We have ZF. Take it up with it and find a contradiction already.

>intense parroting
If you really care about this you should rather pick up your pencils.

>>12511516
>you can't just make numbers up
lol

>>12511294
>294▶>>12511304 >>12511363 >>12511381 >>12511471 >>12511489 >>12511493 >>12511496 >>12511533 >>12511652 >>12511682 >>12511684 >>12511793 >>12511891
> explain to me what the "real" numbers are without using schizo infinities
They are just normal numbers with decimals you can mark exactly on a line...

DUMBFUCK.

Being schiz'd out it magical and fantastically terrifying at the same time. It's literally the crucible of human cognition.

>>12512274
have disproved god yet?

>>12511578
There is evidence for God but there is no evidence for the real numbers.

>>12511853
>this comes from finite natural numbers
How so? Give a finite definition of sqrt(2) then from the finite natural numbers.

>>12512260
>Bounded-from-above set has a least upper bound.
First time I'm seeing this "definition". Least upper bound in what? In the rationals? Presumably you want to quotient out by some equivalence relation. What is it? Is {0,1}={1} in your system
>D) Real number is the rational numbers less than it.
Circular definition. To compare a rational number with a real number, you must have already constructed the real and an order on them.

>>12512317
Please mark e+pi exactly on a line for me and show me.

>>12512474
>>12512477
>>12512480
literal autist

>>12511810
>>12511793
Cauchy sequences and dedekind cuts are memes. They don't work.

>>12512659
they work like a charm

>>12512673
Then clearly you haven't studied enough foundations. Pick your favorite definition and write it down for me so I can point out its flaws.

>>12512680
An acre is the area of a rectangle whose length is one furlong and whose width is one chain.

>>12511294
this is first year first class stuff anon, cant you remember 16 axioms

>>12513768
And how do you know an object satisfying those axioms exists?

you can't put all real numbers on gridline, so it is not ordered set
3<3,5 is meaningful expression, because for scale 0,5 3,5 comes after 3 on the grid
3<pi has no meaning, because there is no such grid that contains both numbers, so you can't say pi comes after 3

>>12511294
what about a grid centered at 3 with step size pi-3

>>12513784
what do you mean by "exists"
i can describe a number by calling it "the number matching the length of the hypotenuse of a right triangle with sides 1 and 1", i can write it down as sqrt(2), does that make sqrt(2) "exist"?
what if i described a number as "the biggest number that's less than 2", and write it down as "x such that for any E, 2-E < x < 2 is true", does that make it "exist"?

>>12511684
define complex numbers

>>12513978
a+bi where a,b are real

>>12513784

because Zeno's Paradox isn't a paradox

of course the infinitely many points of a monotonic (and bounded) sequence are just imaginary labels describing a bounded interval

because an object with velocity can traverse the bounded distance, we know that the endpoint of the interval exists in the real world, hence, its a Real Number.

>>12511294
>schizo infinities
You have it backward, retard. Reals are any number that doesn't fall into a magical infinite pattern, or that isn't frozen magically in time. So they are literally everything you will ever encounter in real life.

>>12511294
>explain to me what the "real" numbers are without using schizo infinities
All the numbers that can be used to measure quantities such as distance, time, mass, energy, velocity, etc.

>>12514278
So... the rational numbers?

>>12514280

>I am going to calibrate my ruler to be comeasurable with the short sides of the 45-45-90 triangle
>what? why can't I measure the long side?
>ok, I will calibrate my ruler to be able to measure the long side
>wtf? I can't measure the short sides now

rational numbers are a meme

>>12514293
Meme or not, they're the only thing we have for that kind of thing.

>>12514300

You could just use a straightedge and compass.

Or just accept that modeling the numbers in between rational numbers with infinite sequences is the only shot you have for distinguishing between pairs of numbers that appear equal due to coarseness of measurement, and pairs of numbers that will always appear equal no matter how fine your measurements become.

>>12511696
Good bait or extraordinary retard.

>>12514312
>modeling the numbers in between rational numbers with infinite sequences
If there was a logically coherent way to do it I'd be all for it. Unfortunately modern mathematics isn't ready for such a thing yet.

>>12512471
So, what evidence is there for the existence of God?

>>12514315

Real Analysis is logically coherent.

Of course, its not possible to prove all true facts about the infinite sequence models of Real Numbers, but you can prove at least some true facts.

>>12514278
Most real numbers can't be computed to arbitrary precision.

>>12514383

While it is true that chaotic systems can challenge the ability of the infinite sequence model (of Reals) to ever improve the precision of the computations, the arbitrary precision property certainly exists for procedures continuous on closed, bounded domains.

>>12512659
From a constructivist point of view, Cauchy sequences are fine if you're careful enough about how you precisely define them. Dedekind cuts are more troublesome, they often require some for of choice and compute poorly. See e.g. http://math.andrej.com/wp-content/uploads/2008/08/abstract-cca2008.pdf

>>12514331
There are no "facts" about anything in real analysis since the propositions don't refer to anything.
And if it were logically coherent then there would be a logically coherent, unambiguous definition of what a real number is. Mathematicians have given up on providing a definition of real numbers long time ago. Turns out it's too hard to do it (while also being logically coherent).
>>12514408
What is your preferred definition of a sequence? I'd love to discuss that with you.

>>12514426

> Real Numbers are the ultimate states of bounded, monotonic sequences
> This definition is coherent because we know it is possible for a particle to completely traverse a bounded distance
> The Bolzano–Weierstrass theorem allows us to pull a bounded and monotonic sequence out of a Cauchy sequence, so the Cauchy sequence has a definable ultimate state

There. Coherent.

>>12514451
>bounded, monotonic sequences
Explain what you mean by the word "sequence".

>>12514473

That is a question for you to think at home.

See me in my office hours for further discussion.

>>12514480
Off the top of my head I can think at least 3 conflicting definitions of what a sequence is.

>>12514489

Just pick the correct one.

>>12514492
I give up. Tell me which one the correct one is.

>>12514495

A sequence is a function defined for all Natural Numbers (and only Natural Numbers).

In the context of defining Real Numbers, the codomain of the sequence is the Rational Numbers.

This definition leaves out finite sequences, but no one questions the ultimate states of finite sequences anyway, so, who cares.

>>12514511
What you did here was delegate the meat of your definition to the notion of a "function". Please explain then what you mean by a "function".

>>12514517

That is an exercise left to the reader.

>>12514518
Off the top of my head I can think at least 3 conflicting definitions of what a function is.

>>12514523

That should have been covered in the prerequisites for this course.

I will need to schedule a meeting about this.

>>12514530
To get a better idea of what you mean by a function, let me ask a couple of questions. Which of the following conform to your notion of a mathematically well-defined function f:N->N:
A) f(n) = n
B) f(n) = {1 if the n'th Turing machine halts and 0 if it doesn't, under your preferred enumeration of Turing machines}
C) f(n)={1 if the Continuum hypothesis is true; 0 if CH is false}
D) f(n)={1 if n is a Godelnumber for a true formula in set theory; 0 otherwise}
E) f(n) = the n'th highest person's on earth height in rounded centimeters
Give reasons why or why not.

>>12511294
the up-to-isomorphism unique completion of the set of rationals with the LUB property.

>>12511294
>>12511363
>>12511381
>>12511435
>>12511471
>>12511489
>>12511493
>>12511496
>>12511516
>>12511533
>>12511559
>>12511578
>>12511605
>>12511640
>>12511646
>>12511650
>>12511652
>>12511654
>>12511682
>>12511684
>>12511689
>>12511695
>>12511696
>>12511699
>>12511700
>>12511705
>>12511786
>>12511793
>>12511802
>>12511809
>>12511851
>>12511853
>>12511891
Engineer here. Who cares? Numbers are numbers and they help us get reasonable approximations to things. Other than that no one gives a fuck. Have sex you fucking retard losers.

>>12514619
>Who cares? Numbers are numbers and they help us get reasonable approximations to things. Other than that no one gives a fuck.
this is what 99% of mathematicians think about this issue, unironically

>>12511294
The numbers between rationals and the rationals.

Metamathematics was a mistake

>>12514774
coping anti-reverse-mathematician

>>12514280
Not just the rationals.

>>12515850
You also include infinitesimals like 1/infinity?

>>12514619
Mathematician here. Who cares? Approximations are approximations and they help us get reasonable understanding of things. Other than that no one gives a fuck. Have sex you fucking retard loser.

>>12511294
diagonal of unit square is inconstructible as a sum of unit lenghts, so it has no lenght, unless we define it as unit lenght

>>12512227
>Now how are you going to check whether your sequence and my sequence
You already defined your sequence to be equivalent. Did that take infinite work?

>>12513807

>>12511294
The smallest complete metric space containing the rationals with the usual distance.

>>12516328
>complete metric space
that is some infinitist bullshit

>>12516345
indeed

explain to me what the "natural" numbers are without using schizo infinities

>pro tip : you cant

>>12516635
summations of unit vector

>>12516635
free monoid on a single generator

>>12511294
>explain to me what the "real" numbers

heckin cute and valid

>>12514619
bitch sex is fuckin easy for you to get because you're gay

>>12516718
Sounds like a porn title.

if reals are baloney then so are complex, analysis and finally physics

now you need to restart everything starting from the from lambda calculus

Brouwer, you have won

>>12511294
completion of the rationals with respect to the usual euclidean metric.

>>12511696
uncountable bait

Should I study the schizo classical mafs to better now what's possible/definable in constructive maths? Such as when a statement can be proved nonconstructive or independent?

>>12514619
>Engineer telling other people to have sex
o i am laffin

>>12511363
You can have a pie and eat it too... you can't eat a pie and have it too...

>>12511294
The real numbers are the set whose elements are completely described by the rational and irrational sets. :) easy

explain to me what reality is without using schizo infinity

guys what's this? Is this on-topic? >>12519036

An infinitist category theory professor and transsexual was teaching a class on David Hilbert, known nonconstructivist.

"Before the class begins, you must get on your knees and worship Georg Cantor and accept that he was the most highly-rigorous being the world has ever known, even greater than Archimedes!"

At this moment, a brave, intuitionist, wildbergian euclidean geometer who had produced 1500 constructive proofs and understood the necessity of algorithmic thinking stood up and held up a 0.999... foot ball.

"How wide is this ball?"

The arrogant professor smirked quite Infinitistly and smugly replied "The equivalence class of the sequence (1,1,1...)"

"Wrong. Only three nines were written down. It there were infinity and real numbers, as you say, are real... then i would need infinite paper."

The professor was visibly shaken, and dropped his japanese chalk and copy of Rudin. He stormed out of the room crying those infinist crocodile tears. The same tears infinitists cry for the “non-measurable sets” (who today live in such luxury that they need not be constructed) when they jealously try to take up space in textbooks from the deserving theorems. There is no doubt that at this point our professor, Cardinal Trannystein, wished he had pulled himself up by his bootstraps and become more than a sophist infinity schizo. He wished so much that he had a gun to shoot himself from embarrassment, but the bullet would take infinite steps to reach his head!

The students applauded and all studied Wittgenstein that day and accepted Kroenecker as their lord and savior. An eagle named “Induction” flew into the room and perched atop the number theory book and shed a tear on the chalk. Wildberger's videos were watched several times, and God himself showed up after descending a finite amount from heaven.

The professor lost his tenure and was fired the next day. He died of gay plague AIDS and was expelled from the paradise Wildberger had created for all eternity.

>>12519178
>dropped his japanese chalk and copy of Rudin

>>12519178
kek

>>12519178
alarmingly based

>>12519178
Infinitists btfo

>>12512474
if figure 1 makes sense then figure 2 makes sense

numbers dont exist you dumbfuck

>>12521050
If 2/5 makes sense then 1/0 makes sense.

>>12521153
false equivalence, by construction x is non-zero

>>12521204
1/0=y
By construction y is nonzero

>>12521216
my figure only assumes that the shortest distance between 2 points is a straight line. therefore by having a triangle of non-zero length sides means the third side must have a length, if you agree that the length of the hypothenuse of the first triangle is 5, then you must also agree than there is a finite length of the second triangle. your construction assumes that devision by 0 is possible i'd like to see you define it first.

>>12521279
>then you must also agree than there is a finite length of the second triangle
I don't.
Your assumption assumes that 2 has a square root. I'd like to see you define it first.

>>12521297
>Your assumption assumes that 2 has a square root. I'd like to see you define it first
no it doesn't, my construction only assumes that a straight line is the shortest distance between two points

>>12521308
And how do you infer that the line has a length?

>>12521315
here is a simple visual proof that the third side has non-zero length greater than 1

>>12521317
Here is a simple visual proof that 1/0 has a non-zero value greater than 1

>>12521321
thank you for your concession

>>12521323
Thank you for your concession that 1/0 is a meaningful number.

numbers arent real, baka

>>12512474
already did you fucking schizo >>12511851

this shit is unironically way worse than the 0.999... = 1 deniers

>>12521153
>If 2/5 makes sense then 1/0 makes sense.
explain your train of thoughts

>>12521493
It's an analogy to illustrate the absurdity of his argument. Just because the hypotenuse of a 3-4 triangle has length, does not mean the hypotenuse of a 1-1 triangle has length.

>>12516199
then what's the reason behind your virginity?

>>12521297
everything has a root, we're working in the algebraic field here