If you could choose a single problem/conjecture...

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You are currently reading a thread in /sci/ - Science & Math

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If you could choose a single problem/conjecture to be solved in mathematics, which one would you choose?

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RH

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[math]\mathsf P \stackrel{?}{=} \mathsf{NP}[/math]

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>>7793063

If P = NP, then obviously N = 1

SOLVED

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>>7793065

>assuming assumptions

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>>7793065

This is literally the most nonfunny meme

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>>7793055

>That picture.

I lol'd.

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>>7793063

Everyone knows they're not.

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>>7793071

"no"

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A correct construction of the reals

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>>7793096

This really.

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>>7793065

every fucking thread

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>>7793096

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>>7793070

This is literally the use of the sum of all natural numbers being -1/12

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>>7793065

hahahahahah so funnay xDDDD

I JUST

I JUST POOPED MY PANTS A LITTLE BIT I LAUGHED SO HARD LOOOOOOOL

HAHAHAHA

Xdddddd

OMMAHGERD LOOOL

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>>7793153

tell me where it is wrong

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>>7793155

The part where you are retarded.

The answer is not one. The true correct answer that is in my paper that just went through peer review and accepted as correct is the hyperreal number 'p-theta' that I invented that is the absolute identity.

Let me explain. 1 is the multiplicative identity, where 1k = k

However, let c denote p-theta. c is the ultimate identity so not only would ck = k but also k + c = 0.

This has a lot of implications when you think about polynomial and non polynomial time when you c time to your algorithm. As it is the absolute identity, if you add c time to your algorithms time, the algorithms time would not changed.

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Can we please just fucking change it to "P ⊂ NP" so retards will stop making that shitty fucking joke?

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>>7793184

I obviously meant k + c = k

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>>7793184

> ck = k but also k + c = 0.

Well both those states existing at the same time is impossible because there is no number that you can use as 'c' that will work for both operations.

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>>7793186

c = 1

k = -1

c*k = k ---> 1*-1 = -1

k+c = k ---> -1+1 = 0

there :^)

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>>7793055

I'd rather change things so that well-established results are no longer true

>adding a "missing mass" term to the number 2*3*5*...*p_n+1 so the primes are finite

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>>7793195

>>7793193

You are not getting it.

My number p-theta is a hyperreal and just like infinity, it does not follow the rules of the reals.

I just defined it to be that way. Think of it like the complex number i that has no real value but we defined it to be the square root of negative one.

p-theta does not have a numerical value. It is more of a logical entity that facilitates the proof that p=np

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>>7793055

Riemann Hypothesis

>or P = NP

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A general solution to Navier-Stokes

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>>7793096

What's wrong with converging sequences of rational numbers?

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Some trivial problem in some trivial subfield, just to troll mathematicians

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>>7793253

You require the Axiom of choice just to define them, which is another way of saying that you never explicitly give the sequence for anything (except a few token examples) but instead just say "if it doesn't exist then my system doesn't make sense, so I'll just wish it into existence!".

The rationals are a subset of the algebraic numbers which are a subset of the definable numbers, which are a subset of the reals. The definable numbers are countably infinite. Every transcendental number you have ever seen is a definable number. Try giving me a sequence to any undefinable number. I claim that at best you will only ever be able to construct sequences for definable numbers and thus will never actually work with the uncountable reals..

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>>7793195

>c*k = k ---> 1*-1 = -1

>k+c = k ---> -1+1 = 0

The well-known cuck equations

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>>7793557

Construction of reals is independent of AC.

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>>7793557

Oh so you want me to define every single irrational number? I fail to understand why.

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>>7793055

Continuum hypothesis

>epic maymays

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>>7793618

Because of

>Muh rigor

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>>7793557

> "if it doesn't exist then my system doesn't make sense, so I'll just wish it into existence!"

Let P be "it doesn't exist"

(P => False) <=> not(not(P => False)) <=> not(P & True) <=> (False || not(P)) <=> not(P)

So if P implies something that is False then P is False.

>>7793627

But that has nothing to do with rigor. Rigor just means you have to make sure that every premises of the theorem is true before assuming its conclusion is true.

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>>7793055

For me, I would like to find closed solutions to the Riemann Zeta function for values of (2n+1) where n is >= 1 and n is an integer.

Either that or the Riemann Hypothesis.

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>>7793651

Wow, you're dumb.

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>>7793737

Great rebuttal, son.

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>>7793757

It's not a rebuttal, just an observation.

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>>7793112

Nice filename

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whether or not 1+1=2, which might not be true if you think about it

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>>7793618

You have to disprove my claim that "you cannot write an undefinable each real using this method and this it doesn't work for all reals". In other words do it for at least one undefinable real.

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>>7793651

>Using the law of the excluded middle in a constructive argument.

What is it like to be such a pleb?

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an explicit form of the hypergeometric function

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>>7793065

You missed a solution, pic very related

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>>7793939

You also can't write down pi, that has nothing to do with constructing it.

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>>7793063

>unconstructive proof that P=NP

>everyone starts looking for the actual algorithm

this would make me kek

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>>7794072

>Can't give a sequence for pi.

>still believes that cauchy sequences are a valid construction of the reals.

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>>7793096

but mah diagonal argument

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>>7793193

it works in the trivial ring, {0} equipped with the typical addition and multiplication

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>>7793603

No it's not anon. Otherwise you can't actually produce your infinite sequences.

With Dedekind cuts you have other problems and the only way to fix them is to introduce AC as well. Though the details are much less obvious.

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>>7794183

http://math.stackexchange.com/questions/488800/dedekind-cuts-for-pi-and-e

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>>7795336

Great (though to be honest those constructions are pretty well known), now give one for an undefinable real.

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0.999...=1

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>>7795427

I never got why this would divide /sci/ so much. I thought whether 0 was a natural number would be more divisive.

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https://www.youtube.com/watch?v=GFLkou8NvJo

maclaurin series for TAU WAU WAU

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>>7794183

I can give you a formula that gives you any digit of Pi. If you want to compute all of them you are welcome to do so.

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>>7795387

Prove me the existence of undefinable reals first please. I don't want to go looking for an element in an empty set.

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>>7795437

Let me explain why it is so divisive.

>90% of /sci/'s users can't join any kind of math conversation that goes beyond high school algebra

>However, they are all really eager to pretend to be smart so they sit down, all day, waiting for a post to be about really simple maths that even they can understand and then they can post about their opinion all day long

>Proofs that 0.999...= 1 are literally high school algebra

>So that big majority all gets pulled in, ready to begin their cries for attention

>Huge fight about who deserves to be right. Again, just to get attention. Probably to brag to their retard friends about how they have SERIOUS mathematical debates online, like th enlightened fedoras they are.

>Thread dies

>Everyone sits down once again waiting for another low level math thread to start

I bet no one even has an opinion. They just want the /sci/ experience without putting in the effort of learning analysis or abstract algebra.

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>>7795437

memes are love memes are life

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>>7795446

WAU = 1 btw lol

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>>7795437

man i got so upset

in the pumping lemma(uvw theorem) it says let x be a natural number and the only solution to the problem in our assignment was with x=0

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>>7795452

It's not divisive, it's le epic trole.

The entire point is that arguments both for and against are based on completely different definitions of .999..., which is exploitable for lulz.

If we require it to be real the nested sphere theorem gives equality

If we define it as the hyperreal 1-h (h is the hyperreal element) then clearly equality doesn't hold.

It's exactly like "is 1 prime?" Depends. Are we defining "prime" as usual to specifically exclude 1 so we can have the fundamental theorem of arithmetic and the results built on it?

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>>7793096

>inplying the reals exist in the first place

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>>7794173

>it is equivalent to AC

This would be glorious.

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>>7795332

You don't know what you're talking about(which isnt surprising as wildberger followers are largely uneducated in mathematics).

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>>7795446

e^2pi = F gave it away

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>>7793253

Choose a programming language and index all Turing machines (or executable program) by [math]i[/math].

Let [math]h(i,s)[/math] be [math]0[/math] or [math]1[/math], depending on whether the machine with index [math]i[/math] halts before [math]s[/math] steps.

For any given pair of numbers, you can indeed compute [math]h(i,s)[/math] by just running the program and wait [math]s[/math] time steps.

Define a sequence of rationals by having the n'th number given by the following sum (where you run through i and run it to step [math] s=n-i [/math])

[math] a_n = \sum_{i+s=n} \frac {1} {2^i} h(i,s) [/math]

This is some number between 0 and 1.

But computing limit n to infinity (and thus s to infinity for all i) requires knowledge whether Turing machines ever halt, which we know to be impossible since the 30's. Thus this real number (insert Wildberger smirk here) is not computable.

And most real numbers are worse, really, because this number is at least definable. Most reals aren't even that.

>>7794173

>>7795619

There's a paper on that idea

http://www.scottaaronson.com/papers/pnp.pdf

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>>7795601

>The entire point is that arguments both for and against are based on completely different definitions of .999..., which is exploitable for lulz.

There is no sane definition of 0.999... besides that of the real number equal to 1.

>If we define it as the hyperreal 1-h

But that makes no sense. The complex numbers, the reals, or some subset of those is the default assumption in any sort of mathematical question. It's like going into a discussion where people are arguing about basic arithmetic and claiming 1+1=0 because "well, technically, in one particular nonstandard number system, this is valid". In the real numbers that everyone uses, 0.999...=1.

>It's exactly like "is 1 prime?" Depends. Are we defining "prime" as usual to specifically exclude 1 so we can have the fundamental theorem of arithmetic and the results built on it?

Exactly. The only sensible way to answer is to assume people are using standard definitions for all the terms involved, and respond "no, 1 isn't prime". Otherwise you become the guy asking "well how are we defining '1'? What about '=' and '+'?"

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>>7795664

???

its just semantics that stat majors like to whack off to

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>>7796242

Why should mathematicians care about computing reals when they just want to complete the rationals?

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>>7795676

Without AC you can only give constructions for definable reals.

>>7795450

I don't claim they exist. I only claim that the definable reals exist, they are countably infinite, and they are a subset of the reals (they satisfy the properties of the reals).

A claim that the reals are uncountably infinite implies that there exist undefinable reals.

A definable real it's defined as follows. Choose a finite alphabet (eg, English+Math, LaTeX, or even the union of all human alphabets plus any symbols you can come up with in your lifetime). Then we say that a word or sentence in the alphabet is a string of symbols (note that we don't distinguish between words and sentences because our alphabet may include a space). The set of all finite words over our alphabet is countable.

>Simply write all the words of length 1 followed by all the words of length 2 followed by all the words of length 3 and so on..

As a corollary we have that the sets of all theorems, definitions, proofs, and so on are countable.

Furthermore, recall that in formal logic we do not allow infinite sentences. Hence if we regard an axiomatic system over a logic as a formal language over an alphabet then the set of all sentences in the axiomatic system is countable. This means that with regards to any mathematical foundation built on logic (e.g. ZFC set theory, where Dedekind cuts and Cauchy sequences live) we have only a countable number of possible sentences. Therefore the number of definable real numbers in such a system must be countable.

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>>7796455

Perhaps they shouldn't, but all the methods for completing the rationals are by construction. They're just bad and handwavy (Wildberger's issue with them).

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>>7793065

also solved if p =0.

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The Grand Riemann hypothesis

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>>7795446

>puh-tolemy

Vi Hart is garbage.

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>>7793055

>using the inferior version

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>>7796933

It's true that there are only countably many definable reals... [math]from \ the \ perspective \ of \ one \ outside \ the \ given \ model \ of \ set \ theory [/math].

But truth is undefinable, and this is why in reality there need not even be a set of definable reals, and even if there were there certainly there need not be a bijection between the set of definable reals and the integers (even though such a bijection certainly exists outside the model). This is how, even if all reals are definable, the reals remain uncountable.

So your claim

>that the reals are uncountably infinite implies that there exist undefinable reals

need not be true.

Furthermore, it isn't true. There exists a model in which all reals are definable (necessarily from outside the model), but in which, of course, the reals are uncountable: choose your favorite transitive model and consider [math]L_\alpha[/math] where [math]\alpha[/math] is the least ordinal such that [math]L_\alpha \models ZF[/math] (and therefore, as Gödel showed, also [math]ZFC[/math] ). One can show that all ordinals in this model are definable (even though, of course, from inside the model there are class-many ordinals), and thus all reals are definable.

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Fermat's Last Theorem. I've been working on it for decades and it to be complete would make the world go round on an or gas mic platter.

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>>7796933

(>>7798245)

And the reals are, of course, uncountable by for example the Cantor diagonalization argument.

But the point is that even though the reals are uncountable, it is possible for all reals to be definable. This is not a contradiction because the bijection exists outside the model, as by Tarski's Theorem on the Undefinability of Truth truth is.. undefinable within the model.

Furthermore, since you take issue with AC, note that the model of set theory [math]L_\alpha[/math] I provided is such that all reals are definable, yet AC still holds.

Moreover, if you are a constructivist (i.e. you believe in the axiom of constructibility), you must also believe in AC, as AC is a consequence of [math]V = L [/math].

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>>7793055

Woodin's [math]\Omega[/math]-conjecture, which, given its definition of large cardinal property, would prove the linearity of the hierarchy of consistency strength.

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>>7793055

fukken saved

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Halting problem.

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>>7798390

Hah.

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>no YM + mass gap

It's as if you don't want to bankrupt the entirety of physics, /sci/.

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>tfw p=np

>tfw potentially cashing in every other problem in mathematics for free

why must the current outlook on things always be so grim

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I wish P was actually NP.

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>>7793137

which is a meme right? I saw a proof that was retarded and the general idea is fucking stupid

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>>7798514

It is a meme, but it's also the correct evaluation of the Ramanujam sum.

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

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The goldbach conjecture

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>>7793819

Noob go read Peano's postulates

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>>7798390

noice m8, fooking nice

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Collatz

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Between every x and x^2 there exists a set of numbers S where

| S ∩ {e | gcd(e, x!) = 1 and gcd(|y - e|, x!) = 1} | > 0

is true for all even y.

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>>7795446

I mean I get the joke.

Its kinda funny

But I feel like its a joke someone in AP calculus in highschool would make.

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>>7793055

giggled at your picture

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1 = 2

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>>7793062

can't decide:

a) generlized RH

b) homotopy hypothesis

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>>7795664

>The probability of selecting any particular number in an infinite set of numbers is 0

no shit.

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traveling salesman problem

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>>7798488

I suspect that an universe where P=NP would not be complex enough to support life.

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>>7795387

lol silly you cant define an undefinable thing.

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