Mathematics - 2021-06-07 (page 2 of 3)

Koro

11:00

etc.

MphLee

@Thorgott @BalarkaSen Ok thank you very much. I go. See you!

Thorgott

cya

Koro

@LeakyNun: I know that a direct (external) product $Z_{n_1}\times Z_{n_2}$ is cyclic iff $n_1$ and $n_2$ are coprime

Any hint which used that?

Leaky Nun

just read what i sent you lol

Koro

I did but please try to understand I don't know that CRT etc

I don't know (yet) may of the words being used there

:'(

@LeakyNun

Lucas Henrique

11:06

the proof isn't that straightforward, you need to use a bit of theory

Koro

It would seem so. I am allowed to use Lagrange's theorem

Lucas Henrique

the classification of (finitely generated) abelian groups uses torsion subgroups. do you know these?

Koro

Torsion?

No

I know only classification of 2p order groups where p is prime

I saw an answer here which gave a hint that: prove that 5 has order 2^{n-2} or something

I don't know how to proceed using the hint also

Lucas Henrique

oh, you need torsion subgroups only for the full classification of when $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic

Koro

I see so I'll wait till I learn these concepts then

Lucas Henrique

11:10

@Koro yes, the case for $n = 2^k$ is pretty much the principle of the CRT with a bit of arithmetic

proving that $U(\mathbb{Z}_{2^n}) \cong \mathbb{Z}_2 \oplus \mathbb{Z}_{2^{n-2}}$ consists of two things:

1) finding an element of $U(\mathbb{Z}_{2^n})$ with order $2^{n-2}$

2) finding an element of $U(\mathbb{Z}_{2^n})$ with order $2$ that does not belong to the first one

3) proof it!

so first prove that thing about 5. that will be your first generator

then find an element of order 2 that is not in the subgroup generated by the first element

since $|U(\mathbb{Z}_{2^n})|=\phi(2^n) = 2^{n-1}$ and the product (as sets) of these subgroups will also be a group and since their intersection is trivial, it will have $2\times 2^{n-2} = 2^{n-1}$ elements. so it will be the whole group. now, they are both normal subgroups since $U(\mathbb{Z}_{2^n})$ is cyclic. thus they are an inner direct sum, which is isomorphic to a external direct sum, etc.

Leaky Nun

@Koro e.g.?

Koro

CRT for example @LeakyNun

Leaky Nun

but theorem 5 didn't say CRT

Lucas Henrique

the whole article might be too much, but that specific theorem @Leaky sent you is just arithmetic

Leaky Nun

you need to learn to read top-down, as my professors tell me

that means, read theorem 5 first, and when (and only when) they reference some lemma, read those lemmas

instead of reading everything from P.1 to P.11

Lucas Henrique

11:22

oops, I said $U(\mathbb{Z}_{2^n})$ is cyclic. sorry. I meant abelian

Koro

@LeakyNun: Some how I was seeing theorem 7 on page 11

I somehow didn't see theorem 5.. I'll take a look. Thanks

@LucasHenrique @LucasHenrique: I am yet to go through your explanation.

A-Level Student

Sorry to repeat myself, but I'd just like to make sure my message stays visible.

Hello everyone. I'm 17, and I've written 2 papers on quite separate topics in maths. I believe the results to be new, but I am seeking a professional opinion about them.
Would any mathematician be willing to have a look at at least 1 of them and comment on it within about a week or two?
They aren't very complicated, so going through them shouldn't take too long.

Thorgott

I'm not the person you are looking for, but it would help in any case if you mentioned the topics of these papers

A-Level Student

@Thorgott I'd prefer not to say the exact topics for fear of someone else figuring out the same results as I have, but I can say that the topics covered are understandable to a strong(ish) high school student.

Edward Evans

Why don't you get one of your teachers to take a look?

porridgemathematics

11:36

hi all, could someone help me understand a sentence from here imgur.com/a/EipKJF7 , 'Roughly speaking, a manifold is a space $X$ that locally looks like $\mathbb{R}^n$ and a submanifold is a subset of $X$ which locally looks like an affine subspace in $\mathbb{R}^n$', what I dont see is why an affine subspace doesn't locally 'look like' the linear subspace its a translation of

Koro

@LucasHenrique Ahh I see, so that's the deal with 5

Thanks a lot @LucasHenrique :)

A-Level Student

@EdwardEvans One of my teachers said it was too complicated for him, but I think he must have been joking; he was probably too busy to have a look at it. The other maths teacher of mine did look at 1 paper, but he isn't a professional mathematician at all: he's done a BsC in maths and sometimes I feel like I know more than him.

porridgemathematics

and if an affine subspace does look like a linear subspace its a translation of (which as far as I can see it should, its even homeomrphic to it by that very translation), isn't the introduction of affine subspaces a little redundant? Can't they have just written 'and a submanifold is a subset of $X$ which locally looks like a subspace of $\mathbb{R}^n$'

Edward Evans

@A-LevelStudent well I still think you should just mention the topics here. Nobody's gonna scoop you lol

Lucas Henrique

@A-LevelStudent you shouldn't be afraid since if simply speaking of which topics it's about makes one figure out the results then it's trivial. :p

Thorgott

11:41

Let me be frank. 1.) If the result is easily replicated just by you giving an indication about it in an online chatroom, it likely isn't worth publishing (unless it's expository in nature, in which case replicability is a non-concern)
2.) It's very unlikely someone will agree to unconditionally proofread something when you're not even willing to disclose what it is. Being understandable to a strong high school student is one thing, but if you want to do serious mathematics, you want the opinion of an actual expert. A high schooler might be able to understand the content, but that is incompa…

Lucas Henrique

I made a paper about combinatorics and generating functions on algebraically closed fields. very vague ideia on what it is about - try telling us

Martin Sleziak

@Jam You don't have to worry about that - if a room is inactive for 14 days, it gets frozen. (If it had low number of messages, it is already after 7 days.)

Thorgott

@porridgemathematics you are correct, affine or not is an insubstantial difference

Martin Sleziak

@Jam Re: "dont know how to invite people". See meta: How do I invite a user to chat?

Thorgott

maybe the authors are doing affine subspaces cause it's more convenient for what they plan on doing, I don't know, but I wouldn't overly concern myself with it

porridgemathematics

11:46

alright, thanks @Thorgott

A-Level Student

I'd rather be on the safe side, thank you though @LucasHenrique and @EdwardEvans . I'll say what areas they're in though:

The 1st paper is number theory.

The 2nd paper is to do with the Fibonacci numbers.

@Thorgott Is that still too vague?

Edward Evans

if you've proved Collatz, you haven't

Lucas Henrique

well, I'm pretty sure less qualified than @Thorgott and, since he isn't your person, neither am I :p

maybe there's some number theorist hanging around...

Edward Evans

hiding

Lucas Henrique

but you're probably gonna want to wait. get to uni, talk to a professor you trust and he'll tell.

not to drag you down but it's really possible that you didn't prove something new. that's a disappointing part of maths. so it's better to wait, get more resources and math culture to refine your results.

A-Level Student

11:58

@LucasHenrique I understand the sense in that, but I thought it'd actually be a good thing for uni's to see if I'd published 2 research papers while still 17, so I would prefer to publish sooner than that.

Lucas Henrique

I see.

A-Level Student

I know what you mean about the disappointment; I've discovered results I thought to be new loads of times and then found out they were already known :)

barista

I'm trying to prove

I tried to combine

CIF and Caratheodory inequality

By CIF, $\frac{f^{(n)}(z_0)}{n!} = \frac{1}{2\pi i}\int_{|z|=R}\frac{f(z)}{(z-z_0)^{n+1}}dz$

But the problem is if I apply B-C inequality to the integrand of CIF,

we take the integral over $|z|=R$ not $|z|=r$ so I can't directly apply the inequality

Any help for this?

Especially I don't understand where that $2^{n+2}$ came from

barista

12:39

So far, I got A((r+R)/2) in the desired inequality

Now I'm doubting if it's a typo...

Lucas Henrique

13:36

every non strict monoidal category is equivalent to a strict monoidal category

lol @ abstract nonsense

Thorgott

isn't a skeleton automatically strict

hmm, I guess it's not entirely clear whether a skeleton inherits a monoidal structure

whatever

Balarka Sen

A-Level Student

@BalarkaSen Why have you posted that here? I don't speak French, but I don't think Tintin has anything to do with Mathematics :) Please refrain from posting things like that.

Balarka Sen

its a skeleton joke

also you can post non-math things here as long as its not spam

A-Level Student

@BalarkaSen Lol, I hadn't realized your intention :) My mistake, no offence intended.

Balarka Sen

13:45

see top right corner for chat guidelines

Edward Evans

rofl

Lucas Henrique

lmao

porridgemathematics

whats the translation?

Balarka Sen

"this... is bizarre"

"where the hell could it have gone?"

porridgemathematics

thanks :) I think i've read the comic this is from before.. good memories

A-Level Student

13:50

From Destination Moon...no clue what the title is in French :)

Lucas Henrique

I love how portuguese inherits latin structure. The comic is barely understandable to me and I haven't ever studied french

Balarka Sen

the story is that thomson and thompson kept seeing each other crossing the other side of the x-ray plate and freaking out about a skeleton haunting the halls

@lucas oh cool

speaking of french

hi @astyx

Astyx

Hey

It's not a coincidence

Balarka Sen

i guessed so

Astyx

Although I do appreciate the Tintin ref

A-Level Student

13:55

I'm going offline now. If anyone agrees to have a look at any of my papers please do use @A-LevelStudent to grab my attention. Thanks, and have fun :)

porridgemathematics

@A-LevelStudent but only professional mathematicians right?

Balarka Sen

i will ask around in my department and get back to you

not many people specializing in fibonacci identities that i know of

porridgemathematics

@A-LevelStudent im starting a phd in september, so im not professional but im on summer holiday and dont mind taking a look at it

of course there are probably way better candidates than me

aarbee

Hi. Without using calculator, can we find the value of sin80?

Balarka Sen

depends

what if you are the calculator

aarbee

13:58

lol

Balarka Sen

you can use a taylor series expansion

sin(80) = sin(90 - 10) = sin(10), then use taylor series for sin(x) around x = 0

i assume 80 means 80 degrees and not 80 radians

Xander Henderson

14:12

@A-LevelStudent In this day and age, you should not fear being "scooped". Post the papers to arXive, or to your own personal website, or to github, or to some other place where you have a timestamp indicating a "publication date". Done. You have asserted priority.

Keep in mind that the goal of academic inquiry is to discover "truth". We collaborate with each other to push human knowledge forward. Trying to keep your results secret is antithetical to that.

A-Level Student

@XanderHenderson If I post them on arXiv, isn't it permanently on there even if it turns out the papers I wrote is utterly trivial? That's why I want a professional opinion about them, to make sure they are worth publishing. Once I have that I fully intend on publishing it on arXiv, so I definitely won;t keep the results secret.

Xander Henderson

@A-LevelStudent What you seem not to be understanding is that professional mathematicians share their work with others all the time.

copper.hat

@A-LevelStudent you can't have it everyway

Xander Henderson

We talk to each other and share ideas.

That is how we push our ideas forward.

Secrecy is antithetical to this.

copper.hat

There is always a risk involved.

Xander Henderson

14:17

Moreover, I highly doubt that someone with a high school background is going to have independently come up with something terribly significant. As an anecdote, I went to high school with an extremely talented and bright young man. As a freshman (in high school), he published a paper which made some progress on a problem which was open at the time (I can't remember which problem---it was either Fermat or Poincare---a complete resolution came a few years later).

Because of the structure of my high school, which was a part of the university system, he had advisors at the local university, and worked with folk at other R1 institutions.

He had a large support network.

Even then, the result he came up with was only a tiny step, and was rendered a dead end a few years later.

Of course, I made the mistake of Google stalking that guy a few years ago---he now works in astrophysics at UC Berkeley, has some named chair, and has over a thousand publications (and is first author or sole author on a significant number of them).

The point is that this guy was brilliant (based on my own personal knowledge of him, as well as his career path), and the best he could do as a high school student was make some very small progress on, and this was done with the help of a large number of experts, who shared ideas openly.

Secrecy is not good for academia. :\

A-Level Student

I see. Still, it won't be a secret for long if someone does have a look at it. I'm also slightly worried that if I publish my papers and they turn out to be trivial (or worse, wrong) then it may affect my reputation if I try to become a professional mathematician. I'm wondering now, is that a silly worry?

Xander Henderson

Oh, I lied. The named position at UC Berkeley was a postdoctoral appointment. He is at University of Arizona now, in a tenure track position (I just Google stalked him again).

leslie townes

i think it is generally understood that your first work might not be your best. also, if it's not published, there's an open question as to whether people would even be able to track it down later. or how much they would care about it.

Xander Henderson

I actually should probably send him an email... I've been toying with the idea of taking some graduate hours in physics so that I can teach physics at my current institution (since we don't actually have any physics faculty here).

leslie townes

lots of people put unpublished notes/etc. on their web pages.

copper.hat

14:24

and in this chat room

Xander Henderson

@A-LevelStudent Nobody cares.

leslie townes

it would be a rare hiring committee that said, we like X, Y, and Z, and W, and R, and S, and T, but we found this thing on a web page from 10 years ago which doesn't seem up to that standard.

Xander Henderson

Like, really. There are probably about 20 people in the world who might be interested in reading my masters thesis (and I have had beers with about half of them). There are maybe 40 or 50 people in the world who might be interested in my phd thesis (and I've had beers with almost all of them).

Nobody cares.

leslie townes

my papers on researchgate get about one download a year. it is probably a web crawler.

A-Level Student

@porridgemathematics thanks so much for offering! I'd really appreciate you having a look at it thanks. How can I contact you?

Thanks for quelling my worry @leslietownes and @XanderHenderson :)

Xander Henderson

14:27

@leslietownes My paper gets four or five downloads every year, but I think a lot of that has to do with the fact that one of my collaborators on that paper is a god damned machine. He cranks out another paper every five or ten minutes.

Most of them are pretty good, too.

leslie townes

those people need to slow down. :)

Xander Henderson

@leslietownes I mean, yes, but only because I need to protect my ego.

leslie townes

it's crazy when people can do that and also be doing good work. more common is the pattern of, each paper in a year has 1/3 of the last paper in it, and 1/3 of the next paper in it, and two parameters vary in the middle. easier to do in some fields than in others.

Xander Henderson

(Honestly, I figured out a long time ago that I am never going to be a superstar, and decided that I can make a much more powerful impact by teaching. I have a few papers that I really, really ought to get out, but haven't found the energy to deal with in the last year. I likely won't do a ton of research beyond that.)

leslie townes

i had an officemate who worked in PDE. there were a lot of people doing that in his field. slightly different boundary conditions or coefficients. turn the crank on the printing press.

i don't think it was professionally useful for people to do that, but they did it anyway.

Xander Henderson

14:31

@leslietownes I think that John got lucky, in that he is working in a field where there is, potentially, a fair amount of low-hanging fruit.

leslie townes

anything applied, it's fairly easy to do too. new data can be a new paper.

Xander Henderson

He studies notions of metric dimension (Assouad dimension, conformal dimensions, lower dimension, etc). While there are a lot of results about the Minkowsi-Bouligand dimension, Hausdorff dimension, and packing dimension, the Assouad dimension (and its relatives) didn't get a lot of attention until the late 90s or early 00s.

leslie townes

again, who knows if it helps. it tends to be pretty obvious when someone is write-only.

i've got a whole lot of stuff ready to go on the Leslie Dimension.

Xander Henderson

And, even then, it wasn't until a few papers came out around 2010 that anyone really started paying attention.

So there are a lot of basic results which need to be proved.

@leslietownes Heh.

copper.hat

worth reading norbert wiener's autobio if you are planning on being a super star

leslie townes

14:33

it sometimes bothered me in grad school that students in fairly new fields seemed to get outside funding more easily, simply because the aspirational stuff (or already proved but not published stuff) they put in their applications sounded rosier than realistic disclosure about work in more established fields.

Xander Henderson

@leslietownes Yeah. cough Applied category theory cough

Oh... I seem to have something caught in my throat...

ahem

More Anonymous

Hey all!

Is there some place on the internet I can ask if a formula I conjectured (now proven) is worth publishing or useful for something? (I've done a physics masters)

Xander Henderson

@MoreAnonymous Not really. This seems more like a question you should put before an advisor or mentor.

Or, really, just write the paper and submit it somewhere. If it is accepted, you have an answer.

But, generally speaking, "a formula" is not publishable.

More Anonymous

@XanderHenderson I don't think I as a physics student could possibly write math upto your level

Xander Henderson

@MoreAnonymous Why not?

leslie townes

14:37

you could post a question on math.SE in the form of "are other versions of this known? do people know of similar results?" with some background on what motivated you to care about it.

More Anonymous

@XanderHenderson Have you seen the math a physicist uses its filled with black magic! :P

Xander Henderson

But if the claim is that a physics student can't write to a reasonable standard, that is an answer, too (it isn't publishable).

More Anonymous

@leslietownes I posted it here

Xander Henderson

@MoreAnonymous No. I don't know the first damn thing about physics.

More Anonymous

13

Q: The Definite Integral Problem (with a twist)?

The Definite Integral Problem (with a twist) In the Riemann integral one essentially calculates the area by splitting the area into $N$ rectangular strips and then taking $N \to \infty$. Here's something I asked myself related to the Riemann integral. Let's say I split the area into say $3$ st...

leslie townes

14:38

oh, you beat me to it.

you stole my good idea, and covered your tracks by going back in time and doing it "first."

Xander Henderson

I took a semester of physics in high school (so I know a very little bit of Newtonian physics, from pre-calculus standpoint), and I took a quarter of quantum mechanics as a graduate student (so, like, I know about certain families of unbounded operators acting on Hilbert space).

More Anonymous

@XanderHenderson Most physicists think there is only one hilbert space

:P

leslie townes

publishability of stuff is pretty context dependent. a lot of journals that focus on expository math, or stuff that might not be on any recognizable research frontier, usually like stuff that ties to something of 'general interest.' 'why should anyone care' is often the threshold question.

there is only one hilbert space.

:)

Xander Henderson

Oh, I also once read an astrophysics paper.

More Anonymous

@leslietownes Some math people have pulled a fast one on me :p

Xander Henderson

14:40

Though, I admit, I only read it because it contained a 1-1 scale figure.

arxiv.org/pdf/1909.11090.pdf

More Anonymous

@XanderHenderson was the error bar +/- 50 km/ sec ?

leslie townes

well, there are finite dimensional ones, and then there's the one true hilbert space.

there is no such thing as an inseparable hilbert space.

Xander Henderson

@leslietownes This.

And then the C-* algebra folk go crazy, with their von Neumann algebras, and Type III factors, and whatnot.

leslie townes

one thing i'm ashamed of is in my dissertation, i put some kind of throaway remark at the beginning like "all X's are assumed separable unless otherwise stated." i never stated otherwise. i should have taken those weasel words out.

More Anonymous

@leslietownes I see :////

leslie townes

14:42

it was about operators on von neumann algebras. :)

Xander Henderson

Oh, and don't forget Temperly-Lieb!

leslie townes

i love type III factors.

speaking of physicists, bob powers (constructor of the first continuum of non isomorphic type III factors) was educated as a physicist. and he was thinking physics when he came up with those.

More Anonymous

@leslietownes I see ... : /

leslie townes

i had the odd experience of asking him to explain an aspect of that construction to me once, without realizing that he both had invented it and was kinda famous within the field for doing so.

sometimes ignorance really is bliss.

Xander Henderson

Heh.

leslie townes

14:47

i never understood the temperley lieb algebra.

more anonymous: do you see?

:)

Xander Henderson

I'll be honest: I don't really get most of that stuff. But we had a couple of guys in our fractals research group at UCR who had a fair amount of expertise in that area (one was a postdoc who had done his phd work under Vaughan Jones; another had done his BS at Vanderbilt, and worked with Vaughan Jones).

More Anonymous

@XanderHenderson I've met Roger Penrose thats the highlight of my academic life

:P

Xander Henderson

So I have sat through a lot of talks on planar algebras and the Temperley Lieb algebra and such. I know some of the big words, but found that most of it went over my head.

More Anonymous

I only know Lie algebra

Xander Henderson

Oh, sh*t... Vaughan Jones died last year.

leslie townes

14:51

yes, that was a surprise to me.

way too soon.

one of the nicest people.

such a new zealander, too. you'd never guess he was a knight of the realm, or whatever he was.

Xander Henderson

I never met him, but his students were very fond of him... I should probably send Mike an email. :\

leslie townes

i didn't know him well but did learn the basics of von neumann algebras from him.

he had a draft of a really good book on the subject, probably still on his webpage. i don't think he ever published it.

"course.pdf" on math.berkeley.edu/~vfr/math20909.html if anyone is curious.

very good notes.

Xander Henderson

15:35

Oh, I just came across a terrible pun:

leslie townes

boo.

Permutator

16:09

@XanderHenderson Jeez, you drink beer?

Drinking beer and smoking is a sin in our culture. Jeez, you guys on weed too?

Xander Henderson

@Permutator I am sorry that you come from a culture which encourages you to pass judgement on those who are not a part of your culture. Personally, I am inclined to live and let live. As long as you aren't hurting others, do what you like.

robjohn

@XanderHenderson I like it, but then I am aberrant in my enjoyment from puns.

Permutator

16:24

@XanderHenderson Your philosophy sounds good but you're aware that you're hurting yourself by derinking beers and bein on weed right?

robjohn

Assuming someone smokes marijuana just because they drink beer is just wrong

Permutator

@robjohn You seem quite knowledgeable in these things. Do you smoke marijuana?

shintuku

something something cardiac benefits wine etc.

porridgemathematics

i dont get the connection between the jehovahs witnessing and the pun post

is there one?

Permutator

@shintuku NO

robjohn

16:26

And there is evidence that drinking moderate amounts of alcohol is not harmful and can even be somewhat beneficial.

@Permutator Never have, not that it's any of your business.

shintuku

also there is evidence the russian monarchy used vodka sales to maintain the serfdom in utter stupidity, how are you to govern without alcohol?

have you seen what happens when you try to prevent people from drinking alcohol?

of course, you might be thinking of the american prohibition

robjohn

@porridgemathematics follow the link, the comment is not about the pun

shintuku

but actually I am thinking of the canadian prohibition, less well known but simultaneous to the american prohibition

quebec almost seceded because of the lack of alcohol

do you understand what this means

porridgemathematics

im dumb whats the link? like the link I click on to open the pun image?

shintuku

now, if you want society to disintegrate, i strongly suggest you force everyone to stop drinking alcohol

they'll all become communists

trust me on this

they almost did because they couldn't watch baseball

imagine what would happen if they didn't have alcohol

robjohn

16:33

@porridgemathematics see the arrow ^^^

porridgemathematics

ah much thanks

oh wow, lol, he must be really passionate about his agenda to link to something that far back

shintuku

the world runs on passion not on fossil fuels, bless their misguided courage in imposing their views on the rest of the world

i see great things for their future

porridgemathematics

haha

Eminem

How do I prove that if A is finite than A is definable?

I know that union of deibale sets are definable, so would a direction of a union n singletongs (where n is the size of A) which are definable are a good start?

Alessandro Codenotti

Definable in which language? Are you allowing parameters?

Eminem

16:45

first order logic

porridgemathematics

say the set is $A = \{b_1,...,b_n \}$, if you're allowing parameters consider the formula $\phi(x,y_1,...,y_n) = (\lor_{i=1}^n x = y_i)$, then $a \in A$ iff $\mathcal{M} \models \phi[a,b_1,...,b_n]$

I think you can only really do it without parameters if your finite set is distinguishable using the functions/relations of your first order language

but I might be wrong

e.g. if your structure is $<\mathbb{N} ; \leq >$ where $\leq$ has the usual interpretation and you want the smallest $n$ natural numbers or something

aarbee

Hi. If square root is shown with \sqrt, how can we show write cube root?

I tried \cubert, didn't work

epsilon-emperor

imgur.com/a/p0gE0Lf

Hello! Could someone help me with my question here?

How do I construct F, given f?

@aarbee See this: tex.stackexchange.com/questions/49043/nice-looking-p-th-roots

porridgemathematics

by the way, on the topic of first order logic, this has been bugging me for a while: in a first order language with equality, and one unary function $f$, with interpretation $f(x) = x^2$ in the structure $<\mathbb{N}, f > = S_1$ and interpretation $f(x) = x^3$ in the structure $<\mathbb{N},f> = S_2$ (so the only difference between these is the interpretation of $f$), are $S_1$ and $S_2$ elementarily equivalent?

if we change $\mathbb{N}$ to $\mathbb{R}$ they are not elementarily equivalent, since for instance $x^2$ has two fixed points is a sentence true in the first structure but not the second ($x^3$ has three fixed points in the second)

but im wondering if we can distinguish between these structures when we change the domain to $\mathbb{N}$ as far as true sentences

aarbee

@epsilon-emperor Thankyou.

porridgemathematics

16:59

for that matter I could have said if we change $\mathbb{N}$ to $\mathbb{Z}$

Transcript for

$Mathematics$ Mathematics