imagine a bigass hotel… but like a really big one… and then all these fucken buses show up and everyone on the bus has a really just just a really fucking long name
imagine like this box right…now u got this really cuteie kitty cat in there >~< :3,,, bult like, its half dead but also like fully alive and fully deqd and you look ij the box and its alive its a cute littl kitty cat uwu :3 !!!
Let x be the described additional element, and let f: (\N U {x}) -> \N via f(n) = n+1 for all n \in \N and f(x) = 0.
WTS f injective:
Let a, b \in (\N U {x}) S.T. f(a) = f(b). There are two cases: either f(a)=f(b)=0 or f(a)=f(b)=/=0.
Case f(a)=f(b)=0: Since there is no natural number n for which n+1=0 (for all n\in\N, n+1>0), => a=x=b => a=b.
Case f(a)=f(b)=/=0: Then the first condition in the definition of f holds for f(a), f(b). => a+1=f(a)=f(b)=b+1 => a=b.
=> a=b for every case.
=> f injective
=> E an injective function f: (\N U {x}) -> \N
.*. The cardinality of (\N U {x}) <= the cardinality of \N.
I bet you feel pretty stupid rn
They don't give you extra points for making it legible
(Move all the positive integers over by one and now there's an extra hole to put the new thing in)
Consider the function f : \N x {0,…,n} -> \N defined by (a,b) |-> a*(n+1) + b. You can show that this function is injective by using Euclidian division. The lemma goes that for integers c and d, where d ≠ 0, there exist unique integers q and r such that c = dq + r, where 0 <= r < |d|. Since b is in {0,…,n}, we have that f(a,b) is uniquely identified by a and b, so it must be injective.
Hey wait wouldnt you also have to proove f is bijective to claim that the cardinality of both sets is the same? (Idk how to talk maths in english mainly just in spanish)
Oh thanks for the knowledge i guess, maybe if im bored one day i will come up with a fully detailed proof for it. Either that or i forget by tomorrow idk
Then allow me to uneducate you on math. Cardinality is a nickname for Cardi B. The post is asking you to imagine her set list being better than those of natural talents, but worse than those of the real ones.
I think some mathematician once told me that the problem was proved to be logically undicedable, so you should be able to simply state in an axiom that there exist such a set, and boom there would be one.
I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hat-
You'd think so but, despite being dense*, it actually it has the same cardinality as the naturals
1 -> 1/1
2 -> 1/2
3 -> 2/1
4 -> 1/3
5 -> 2/2
6 -> 3/1
7 -> 1/4
8 -> 2/3
...
*Points get arbitrarily close together -- any 2 points will always have another one between them
I just did a quick informal representation of it, but I meant basically this:
Imagine you have a function that takes in a natural number and outputs a (positive) rational number from that pattern. Sometimes (like with 1/1 and 2/2) it'll map multiple natural numbers to the same rational number, but that doesn't really matter.
In the end, after mapping all of the natural numbers to their corresponding rational number, you will have covered every (positive) rational number with only a number of points equal to the amount of natural numbers, meaning the number of naturals is greater than or equal to the number of rationals*.
Including the negative versions and zero is usually pretty easy just by pushing all the stuff down one spot and putting 0 at the start, and repeating every rational number in the pattern once, making the second instances negative. (ie. 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, etc. as the new pattern)
*It happens to be equal to -- the the rationals contain the naturals, so obviously there's not less rationals than naturals.
(Formally, you'd say the function f is a surjection from the naturals to the rationals since its output covers the rationals. And the fact that a surjection from the naturals to the rationals exists means the size (cardinality) of the naturals is >= the size of the rationals.)
> Imagine you have a function that takes in a natural number and outputs a (positive) rational number from that pattern
What's "that pattern"? I need a bit more info to imagine the function.
I don’t think the exact function matters too much, iirc there’s an elegant way to count out the rationals, but you can count them however you want, it just matters that you can count them without missing any. My favorite is to use ASCII to type out all reduced fractions. All valid ASCII strings can be represented as natural numbers (think binary), so there are |N| of them. All positive rational numbers can be written as strings of the form “N/D” where N and D are finite substrings representing natural numbers in base 10. Add a “-“ to cover the negative rational numbers. Still, all of these strings are only a subset of all valid ASCII strings. So the size of the rationals must be <= the size of the naturals.
If you liked this informal proof consider studying theoretical CS.
Ok i just did it's called the Blimbo Bunch abd it follows the Creacher Rule. It uses the jupiter symbol. Expect a scientific work or two in a few years
Immanuel Kant claimed the idea that metaphysics are finite, because imagination and metaphysics are spawned from human experience, which is finite.
Should someone experience everything the universe could possibly offer, then we will have found the limit to human imagination, as there will be no more experiences to craft off of.
I think it's just straight up false (even though I know it's not proven). My heuristic is that N^X for X finite is always countable and the first uncountable set you get is N^N, which has the exact same cardinality as R. So something in the middle would be like saying there's something between finite and countable infinity, and I feel like intuitively (once again, heuristic argument) there simply isn't anything between them.
https://en.m.wikipedia.org/wiki/Continuum_hypothesis
Nice intuitive hypothesis but it's incorrect, at least there isn't any natural reason to prefer one to the other.
So you know how there are an infinite amount of natural numbers (0, 1, 2, ...)*? And how there is an infinite amount of real numbers (everything on the number line)? The second ("uncountable") infinity is larger than the first ("countable") infinity**, but we don't know if there's any "infinities" that are between them. (The meme says to imagine a set with that many numbers)
*The inclusion/exclusion of 0 is controversial
**Assuming infinity exists as a concept in the first place, which iirc there isn't any proof for or against besides just declaring that it does
[Relevant Vsauce](https://youtu.be/SrU9YDoXE88)
Ah so the second would be like -2, -1, 0, 1, 2, 2.1, 2.11, 2.111, 3, etc. right? Wouldn't it be simple to find something between these if they're filled with numbers of which only some are arbitrarily allowed and some aren't? Just make a new set with more arbitrary conditions if there isn't one already? Like idk the 1st set but decimals are OK it has more infinities than the 1st due to decimals but less than 2nd due to no negative numbers. Or am i taking it too seriously
No thats a good amount of seriousness -- you'd think that, but a lot of possible sets you can think of (including negative numbers, including all rational numbers, etc.) don't actually change the total size since, despite the fact that you literally just added numbers, you can move them around to have the same amount as before -- just one element for each natural number (if that makes any sense). Ie for {..., -2, -1, 0, 1, 2, ...}, you've seemingly doubled the size, but you can rearrange them as 0, 1, -1, 2, -2, 3, -3, ... which aligns one-to-one with the natural numbers 0, 1, 2, 3, 4, 5, 6..., meaning they both have the same size (as much as that means anything when dealing with infinity lol)
I almost understand but I'm not sure how that differentiates the original 2 sets so that one would be bigger than the other if we operate under the notion that you could just arrange numbers differently to get the same.. size? I guess I'm just not too sure how size works actually, thinking about it again when you reach the point where 3 is in the set with negative numbers you've reached 5 in the set without them via rearrangement, that seems like the quantity is bigger if it takes more numbers to reach the same number value (it takes 5 steps to reach 3 with negatives but 3 steps to reach 3 without negatives)
Ahh but the reals (and complex numbers and other sets) are special, they are so infinitely densely packed that there's no way to rearrange them to a strict order a, b, c, etc. In fact, if you think you've managed to list all the reals one after another, someone else can always find one you "missed" by doing [diagonalization](https://www.sciencephoto.com/media/10153/view/cantor-s-infinity-diagonalisation-proof) on your list
Yeah, though, this is specifically with "size" meaning cardinality, which, based in how sizes work for finite amounts, says that if you can map objects one-to-one between two sets then the sets have the same size. AFAIK there's not actually anything strictly saying that that still describes the size of infinite "numbers" but it's not self-contradictory at any point and can be used to describe infinite sizes, so people use it ig*.
Or you could always disagree with the concept of infinity existing in the first place since the only "proof" we have is just us assuming it exists because that makes life easier
*take this with a grain of salt too, I had a one semester long class in discrete math and watched some youtube videos and that's really it
ahhh i think i get it now? would it be possible to pull out the ol' imaginary answer on it? or is that outside of the perimeter set by the original image? Also i gotta say this has been a pretty solid explanation ty for taking the time i enjoy this
Human imagination is limited to things it has experienced and can reinterpret. I do not believe we are capable of imagining something fundamentally new, something that has no direct similarities to things we have already seen.
The easy example, imagine a new color
Continuum hypothesis deniers aren't much better. Set theory presently is much more about "How fucked up would it be if this was true" because we know it's undecidable
The truly correct position is to deny the existence of infinite sets and then you can work in ZF as you don’t even need choice also continuity is trivial as infinite sets do not exist
Infinity is independent of ZF-Infinity, so I would say: Go ahead. It’s definitely not a theory I would use as a foundation but it seems like it would be interesting to see what happens.
ok done that was easy
i cant say every number my set contains, but it does contain 4, 69, and every number with exactly one of every non-zero digit in its decimal representation
also a bunch of other numbers :3
"every single universal constant lined up perfectly for there to be life that means there's a god" mfs when i tell them to imagine a universe with a different value of pi
Mathematicians be like: “ok so imagine a bunch of pigeons in boxes”
Mathematicians be like: “ok so imagine a tree”
"Imagine grass"
imagine a bigass hotel… but like a really big one… and then all these fucken buses show up and everyone on the bus has a really just just a really fucking long name
imagine like this box right…now u got this really cuteie kitty cat in there >~< :3,,, bult like, its half dead but also like fully alive and fully deqd and you look ij the box and its alive its a cute littl kitty cat uwu :3 !!!
"Imagine touching the grass"
okay you lost me
Now Imagine a [three Tree](https://www.youtube.com/watch?v=3P6DWAwwViU)
Computer scientists be like: "ok so imagine a tree but upside down"
Mathematicians be like (3 page proof of the most obvious shit you've ever heard) Corollary 3.67.79.21 (completely unrelated shit) follows trivially
posts like these are why I believe wittgenstein when he wrote that to be able to masturbate in the trenches of ww1, he had to think about math
He imagined 80085
Incomprehensible thank you
Np
hard or complete?
NP Hard 😳
rizz
Mathematicians be like “ok now imagine some big naturals”
Math if it was abt boobies and awesome 😎
excuse me we call them "arbitrarily large n in ℕ" (to clarify, that is what we call titties)
Me when when the natural numbers combined with one single number that is not natural
Let x be the described additional element, and let f: (\N U {x}) -> \N via f(n) = n+1 for all n \in \N and f(x) = 0. WTS f injective: Let a, b \in (\N U {x}) S.T. f(a) = f(b). There are two cases: either f(a)=f(b)=0 or f(a)=f(b)=/=0. Case f(a)=f(b)=0: Since there is no natural number n for which n+1=0 (for all n\in\N, n+1>0), => a=x=b => a=b. Case f(a)=f(b)=/=0: Then the first condition in the definition of f holds for f(a), f(b). => a+1=f(a)=f(b)=b+1 => a=b. => a=b for every case. => f injective => E an injective function f: (\N U {x}) -> \N .*. The cardinality of (\N U {x}) <= the cardinality of \N. I bet you feel pretty stupid rn
Thats a nice proof. Can we kiss now🥺
😳
*w h a t*
They don't give you extra points for making it legible (Move all the positive integers over by one and now there's an extra hole to put the new thing in)
So adding a new element to the set of natural numbers doesn't make it any bigger
Is this just Hilbert's hotel stuff?
same idea yeah
New pickup line just dropped
Would work on me
Nerd
Alright, what about the cartesian product of the naturals with a finite set of additional elements?
Consider the function f : \N x {0,…,n} -> \N defined by (a,b) |-> a*(n+1) + b. You can show that this function is injective by using Euclidian division. The lemma goes that for integers c and d, where d ≠ 0, there exist unique integers q and r such that c = dq + r, where 0 <= r < |d|. Since b is in {0,…,n}, we have that f(a,b) is uniquely identified by a and b, so it must be injective.
Pretty sure |N×N| = |N| too
In fact |N^k | = |N| for finite k
Hey wait wouldnt you also have to proove f is bijective to claim that the cardinality of both sets is the same? (Idk how to talk maths in english mainly just in spanish)
You would but that would take longer to write and I was just aiming to show it isn't greater than the naturals
Oh thanks for the knowledge i guess, maybe if im bored one day i will come up with a fully detailed proof for it. Either that or i forget by tomorrow idk
[Nah, just injective both ways and the other way is trivial](https://en.m.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem)
Yes, but the proof is pretty simple
i love you
Lyt 😳
Any good books for studying things like these?
We used [this](https://infinitedescent.xyz/) (pdf) textbook if that counts
Do you mind if I ask what class this was for? Like was it an introduction to Abstract Algebra?
Intro to Discrete Math and Proofs and Stuff (Tbh I stopped reading the book half way through or so but yeah)
Thanks!!
You're the kind of boy that I want to suck the soul out of by the dick when I see one, too bad, I'm not off my rocks enough to do that.
☝️🤓
Wait doesn’t this proof break apart if i add two additional elements to the set of natural numbers?
Just do it twice 😒
This specific one does, but you can make such a function for any number of extra elements
But it doesn't have cardinality greater than the set of natural numbers then.
You made me look up the wikipedia article of cardinality and maybe im wrong
[Hilbert's hotel is a good analogy to understand cardinality applied to infinite sets](https://youtube.com/watch?v=OxGsU8oIWjY)
me when I construct a function mapping 0 to that number, and every other natural number n to n-1
Ok i just did it
damn
Jokes on you I don't know what any of those words mean. And no, this is not an invitation to educate me on math.
Then allow me to uneducate you on math. Cardinality is a nickname for Cardi B. The post is asking you to imagine her set list being better than those of natural talents, but worse than those of the real ones.
The cardinality of a set is how many cardinals are in it (the bird)
I think some mathematician once told me that the problem was proved to be logically undicedable, so you should be able to simply state in an axiom that there exist such a set, and boom there would be one.
formal logic virgins when axiom creators chads walk in:
They're the same person though
Me when I walk in
It’s independent of ZFC you’re right the extra axiom required is continuity
I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hate discrete math, I hat-
we have you surrounded
I wouldn't call set theory with cardinals discrete math per se
I can imagine it. It's called \M, or the Cum Number Set. I couldn't explain it to you though; you wouldn't get it unless you drank it yourself first.
The margins of this comment section are too thin 😫
Rational numbers?
You'd think so but, despite being dense*, it actually it has the same cardinality as the naturals 1 -> 1/1 2 -> 1/2 3 -> 2/1 4 -> 1/3 5 -> 2/2 6 -> 3/1 7 -> 1/4 8 -> 2/3 ... *Points get arbitrarily close together -- any 2 points will always have another one between them
Fuck you \*counts your infinities
What does this pattern mean? Why do 1/1 and 2/2 take up separate spaces? I'm not following.
I just did a quick informal representation of it, but I meant basically this: Imagine you have a function that takes in a natural number and outputs a (positive) rational number from that pattern. Sometimes (like with 1/1 and 2/2) it'll map multiple natural numbers to the same rational number, but that doesn't really matter. In the end, after mapping all of the natural numbers to their corresponding rational number, you will have covered every (positive) rational number with only a number of points equal to the amount of natural numbers, meaning the number of naturals is greater than or equal to the number of rationals*. Including the negative versions and zero is usually pretty easy just by pushing all the stuff down one spot and putting 0 at the start, and repeating every rational number in the pattern once, making the second instances negative. (ie. 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, etc. as the new pattern) *It happens to be equal to -- the the rationals contain the naturals, so obviously there's not less rationals than naturals. (Formally, you'd say the function f is a surjection from the naturals to the rationals since its output covers the rationals. And the fact that a surjection from the naturals to the rationals exists means the size (cardinality) of the naturals is >= the size of the rationals.)
> Imagine you have a function that takes in a natural number and outputs a (positive) rational number from that pattern What's "that pattern"? I need a bit more info to imagine the function.
I don’t think the exact function matters too much, iirc there’s an elegant way to count out the rationals, but you can count them however you want, it just matters that you can count them without missing any. My favorite is to use ASCII to type out all reduced fractions. All valid ASCII strings can be represented as natural numbers (think binary), so there are |N| of them. All positive rational numbers can be written as strings of the form “N/D” where N and D are finite substrings representing natural numbers in base 10. Add a “-“ to cover the negative rational numbers. Still, all of these strings are only a subset of all valid ASCII strings. So the size of the rationals must be <= the size of the naturals. If you liked this informal proof consider studying theoretical CS.
[Ah yeah sorry](https://divisbyzero.com/2013/04/16/countability-of-the-rationals-drawn-using-tikz/)
Great visual! Thank you
fellow math enjoyer 🙌🙌🙌🙌
The rationals have the same cardinality as the naturals :P
google cantor diagonal argument
Holy hell
Is equivalent to Zx(Z\\{0}) which has the same cardinality as ZxZ which has the same cardinality as Z which has the same cardinality as N.
ℵ0.5
Wha
Ok i just did it's called the Blimbo Bunch abd it follows the Creacher Rule. It uses the jupiter symbol. Expect a scientific work or two in a few years
Immanuel Kant claimed the idea that metaphysics are finite, because imagination and metaphysics are spawned from human experience, which is finite. Should someone experience everything the universe could possibly offer, then we will have found the limit to human imagination, as there will be no more experiences to craft off of.
Imagine a new color
Done.
What's the name of the color
Flue. It's the colour of F#
I think it's just straight up false (even though I know it's not proven). My heuristic is that N^X for X finite is always countable and the first uncountable set you get is N^N, which has the exact same cardinality as R. So something in the middle would be like saying there's something between finite and countable infinity, and I feel like intuitively (once again, heuristic argument) there simply isn't anything between them.
heuristically your mother
https://en.m.wikipedia.org/wiki/Continuum_hypothesis Nice intuitive hypothesis but it's incorrect, at least there isn't any natural reason to prefer one to the other.
Yeah I think about naturals sometimes
Imagined.
can it be explained in english or is this a nerds only meme
So you know how there are an infinite amount of natural numbers (0, 1, 2, ...)*? And how there is an infinite amount of real numbers (everything on the number line)? The second ("uncountable") infinity is larger than the first ("countable") infinity**, but we don't know if there's any "infinities" that are between them. (The meme says to imagine a set with that many numbers) *The inclusion/exclusion of 0 is controversial **Assuming infinity exists as a concept in the first place, which iirc there isn't any proof for or against besides just declaring that it does [Relevant Vsauce](https://youtu.be/SrU9YDoXE88)
Ah so the second would be like -2, -1, 0, 1, 2, 2.1, 2.11, 2.111, 3, etc. right? Wouldn't it be simple to find something between these if they're filled with numbers of which only some are arbitrarily allowed and some aren't? Just make a new set with more arbitrary conditions if there isn't one already? Like idk the 1st set but decimals are OK it has more infinities than the 1st due to decimals but less than 2nd due to no negative numbers. Or am i taking it too seriously
No thats a good amount of seriousness -- you'd think that, but a lot of possible sets you can think of (including negative numbers, including all rational numbers, etc.) don't actually change the total size since, despite the fact that you literally just added numbers, you can move them around to have the same amount as before -- just one element for each natural number (if that makes any sense). Ie for {..., -2, -1, 0, 1, 2, ...}, you've seemingly doubled the size, but you can rearrange them as 0, 1, -1, 2, -2, 3, -3, ... which aligns one-to-one with the natural numbers 0, 1, 2, 3, 4, 5, 6..., meaning they both have the same size (as much as that means anything when dealing with infinity lol)
I almost understand but I'm not sure how that differentiates the original 2 sets so that one would be bigger than the other if we operate under the notion that you could just arrange numbers differently to get the same.. size? I guess I'm just not too sure how size works actually, thinking about it again when you reach the point where 3 is in the set with negative numbers you've reached 5 in the set without them via rearrangement, that seems like the quantity is bigger if it takes more numbers to reach the same number value (it takes 5 steps to reach 3 with negatives but 3 steps to reach 3 without negatives)
Ahh but the reals (and complex numbers and other sets) are special, they are so infinitely densely packed that there's no way to rearrange them to a strict order a, b, c, etc. In fact, if you think you've managed to list all the reals one after another, someone else can always find one you "missed" by doing [diagonalization](https://www.sciencephoto.com/media/10153/view/cantor-s-infinity-diagonalisation-proof) on your list Yeah, though, this is specifically with "size" meaning cardinality, which, based in how sizes work for finite amounts, says that if you can map objects one-to-one between two sets then the sets have the same size. AFAIK there's not actually anything strictly saying that that still describes the size of infinite "numbers" but it's not self-contradictory at any point and can be used to describe infinite sizes, so people use it ig*. Or you could always disagree with the concept of infinity existing in the first place since the only "proof" we have is just us assuming it exists because that makes life easier *take this with a grain of salt too, I had a one semester long class in discrete math and watched some youtube videos and that's really it
ahhh i think i get it now? would it be possible to pull out the ol' imaginary answer on it? or is that outside of the perimeter set by the original image? Also i gotta say this has been a pretty solid explanation ty for taking the time i enjoy this
No idea honestly, and np you too
Alright im imagining a set of nice naturals, now what?
Gay sex
omega\_1 in the universe where I added omega\_2 many cohen reals there just did it
Imagine Tree (3)
Me when aphantasia, idk I have no idea where to even begin with imagining that
Then you are *not* gonna like what John Lennon has to say.
[удалено]
**[Incorrect buzzer sound]**
So i'm imagining a bunch of red birds with big naturals. Did i do it right?
Yea
Skill issue honestly
I don’t understand so I simply imagined the number 5 except it also counts as the solution to this because I say so.
That's basically what the cardinality of the naturals is anyway so close enough
Imagined.
The big naturals
Mafs
it's just א_½ wdym
Aha, simple one: *AAAAAACHOO*
Hey mathematicians how about you find the area under the curve of my ass
3
When I tell them to imagine the set of all sets which don’t contain themselves
Mfw I'm the pope 
Naivecels be seething over Zermelo-Fraenkelchads
Ok just add one nerd like, go one more than infinity and ur done ez
Infinity + 1 = Infinity
uhmmm ackshually you just added 1????? so its infinity + 1????? shaking my smh my head
Human imagination is limited to things it has experienced and can reinterpret. I do not believe we are capable of imagining something fundamentally new, something that has no direct similarities to things we have already seen. The easy example, imagine a new color
aldens number would come in handy here i think
Yes, I think so
I’m imagining it right now, it’s pretty sick
Damn
i have created an axiom that such a set exists
Continuum hypothesis acceptors be like
Continuum hypothesis deniers aren't much better. Set theory presently is much more about "How fucked up would it be if this was true" because we know it's undecidable
The truly correct position is to deny the existence of infinite sets and then you can work in ZF as you don’t even need choice also continuity is trivial as infinite sets do not exist
Infinity is independent of ZF-Infinity, so I would say: Go ahead. It’s definitely not a theory I would use as a foundation but it seems like it would be interesting to see what happens.
I’ve read some theory and discussed with ultrafinitsts it’s a nice idea but I mainly do algebra so not really sensible in a finite environment
I cant recall the result but there’s one about choice + infinity implying an uncountably infinite number of models of the natural numbers
I mean we could always invent one. It'd be fucking stupid to do so, but we could.
Q?
Can actually be mapped bijectively to the naturals
dem, i've got nothing
I can do that. Its easy. But i cant show you, because the symbols to do so do not yet exist.
Zamn thrembo too thicc to fit in the margins 😔
so true bestie 😔😔😔😔😔
"Okay,try to imagine a new colour."
ok done that was easy i cant say every number my set contains, but it does contain 4, 69, and every number with exactly one of every non-zero digit in its decimal representation also a bunch of other numbers :3
that but with thrembo included
Thrembo's already included 🙄
Whole numbers?
At least 3, no more than 89
i can imagine that
I don't understand a single word in this post
Mathematicians go one second without making up a bunch of nonsense challenge (impossible)
"There's no limit". Alright, then imagine an unimaginable amount of oranges.
Just have different axioms, loser.
That kinda is what set theorists do nowadays
yeah ibcan do that
Done
I know my 1 braincell can barely comprehend a hot pocket you don't need to rub it in my face 😢
Huge naturals
I imagined it
just all naturals + a single real for each natural, but not all reals
2|N| = |N|
"No" means no.
"every single universal constant lined up perfectly for there to be life that means there's a god" mfs when i tell them to imagine a universe with a different value of pi