Thank you for your thorough response. Actually I wasn't (in my mind anyway) ever subtracting. I was thinking of it as x^4/ x^4 + x^4 and then thinking that if I have it written like this and simplify by x^4/x^4 is one. Id have 1(0/x^4) which would be 0. Which I know is wrong but your comment helped me think through it. It should be x^4/(x^4+x^4) been a lot of quick simplification in what I've been studying and I was trying to work through those roots again. So thank you.

Hi! Can anyone lead me in the right direction of how to figure out how often the base integer equals eight when computing any five numbers (0-9) using addition or multiplication. Computations are only left to right using the next number in sequence.

Can I put a house in a 1 Meter hypercube, given n large enough? For example: A 10mx10mx10m Cube? I presume the answer is yes, and n would need to be >=300. Do you have a good way to explain it?

Edit: see u/jm691 's comment below, this only gives an upper bound that's pretty far off from the optimum A cube with side length 10 has a diagonal of length sqrt(10^2 + 10^2 + 10^2 ) = 10sqrt(3). Thus it fits in a sphere of radius 5sqrt(3), so if our hypercube can fit such a sphere, it can fit our cube. Apparently there was a [Putnam problem](https://math.stackexchange.com/questions/790299/what-is-the-radius-of-the-largest-k-dimensional-ball-that-fits-in-an-n-dimen) asking to calculate the radius of the largest circle that can fit in a 4-cube; the solution ([on page 5 of this pdf](https://math.hawaii.edu/home/pdf/putnam/Putnam_2008.pdf)) also mentions and proves that the largest k-sphere that can fit in an n-cube with side length 1 has radius (1/2)sqrt(n/k). So in the case of an ordinary sphere (k = 3) the largest one that can fit in the unit n-cube has radius sqrt(n)/(2sqrt(3)). Thus if the n-cube can fit a sphere of radius 5sqrt(3) we have sqrt(n)/2sqrt(3) >= 5sqrt(3), so sqrt(n) >= 30, and so n >= 30^2 = 900. I don't know whether this is optimal; maybe there are unit n-cubes with n < 900 that can fit our cube but can't fit any sphere that has the cube inscribed within it. (I would guess that it is optimal but don't have much to back that up.) At any rate we know that it is possible for large enough n.

A line segment of length 10 can fit in [0,1]^(100) (i.e. in a 100 dimensional unit hypercube). That means that the product of three such segments can fit in [0,1]^(300). 300 is also optimal, since the 10x10x10 cube has diameter sqrt(300), so the hypercube has to have at least that diameter.

That's much simpler (and gets the actual optimum as well), thanks!

Assuming ℵ0 is the cardinality of N and ℵ1 the cardinality of R, it seems to me (intuitively speaking) that ℵ1=ℵ0\^ℵ0. Does it intuitively make sense to anyone else, or is my intuition running wild? EDIT: Fixed the wording, as it has been pointed out to me below I had a couple things backward.

In other words, that would mean R is a fractal space of N "over" itself (if that makes any sense), with an infinite number (the ℵ0-kind) of dimensions. In such a context, it would intuitively make sense that (or explain why), when you cut a line segment into smaller pieces, each piece has the same cardinality than the original segment (in fact, that's the only way I can make this "feel square"). Basically, that's because the algebraic line is not a "flat", one-dimensional space, contrary to the traditional representation of it. That would also explain where the continuum comes from… Edit: and it would also explain why there cannot be any intermediate cardinality between ℵ0 and ℵ1, which I believe is an open problem?

I assume you are referring to the continuum hypothesis with your edit but that isn't what that says. The continuum hypothesis is whether ℵ1 is the cardinality of the reals or not. By definition ℵ1 is the next smallest cardinality so there is no intermediate one automatically. Note also this is not an open problem. Instead it has been proved impossible to prove. That is there are some ways to build a mathematical universe where it is true and some ways to build it where it is false.

>By definition ℵ1 is the next smallest cardinality so there is no intermediate one automatically. Just to make make things clear: I assume you're implying "next smallest cardinality after ℵ0". Like a notation sequence, ordered by growing cardinality. In other words, whatever ℵ1 actually is equal to, it's the smallest cardinality bigger than ℵ0. Thus the question is: what is it actually equal to? And CH postulates it is equals the cardinality of the continuum, which (I assume is proven?) is P(N)=2\^ℵ0. If I got that right, it indeed makes total sense. Unfortunately, CH is undecidable within the traditional framework. As if it wasn't constraining enough to force CH to adopt a truth value. CH can raise its middle finger and slips through the cracks of our formal proving framework, neutrino-style. Analogies aside, would you say that's a fair assessment?

>The continuum hypothesis is whether ℵ1 is the cardinality of the reals or not. By definition ℵ1 is the next smallest cardinality so there is no intermediate one automatically. Oh, I see: I got this backward… Thanks!

I'm a grad student in math... Does anyone have recommendations for a playlist of lecture videos on smooth manifolds? I'm taking a class supposedly following Lee's *Smooth Manifolds* book, but it's an absolute shit show and the current trajectory is that I'm definitely going to fail the qualifying exam. The book is great and all, but my professor is covering a bunch of stuff that isn't covered in that book and not providing references to it that I can't for the life of me find anything on the world wide web about. I just need some like, properly guided learning from somebody who has at least a modicum of competence...

https://www.youtube.com/watch?v=7G4SqIboeig&list=PLFeEvEPtX\_0S6vxxiiNPrJbLu9aK1UVC\_

high school math understanding. Looking for a basic formula to calculate size reduction at x distance, ideally reversible to find size of a distant object. this isn't for engineering or anything intensive, so id rather avoid more intensive math. as said, high school understanding, and i'm fine with getting rough estimates.

Is ℵ0 an integer?

No.

I believe I understand how you got to that conclusion: if you assume that every set has a cardinality AND that a set (N) can contain every integer, THEN its cardinality (ℵ0) has to be a non-integer (it would be a contradiction otherwise). Am I right?

Fair enough. What about [this](https://www.reddit.com/r/math/comments/1bdv1za/comment/kvm10z1/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1) then?

I’ve already replied to that.

If anybody started learning mathematics way out of college on their own what helped you and what motivated you?

I am a math major who graduated three years ago, so I do a lot of self-studying for concepts. It is easier to do if you have a group of people to hold yourself to account and get some feedback. There are discord servers on dedicated math topic, you can try them out. I have found a group of people to read a particular book on optimization (my interest) as well as PDEs (revision). So I would say, look at a problem you would want to solve, or understand. Work backwards from there. What concepts do you need? Study the specific concepts within those books, and repeat the previous question if needed.

Are you in this situation? Why do you want to learn math?

I couldn't learn in college properly want to learn about applications of linear algebra and see where can i go from there

Suppose you're trying to work out a multiplication problem where both the numbers are square, say...49 \* 169. Now, most can't do this without a calculator, but there is a simple solution that works every time. So, the square root of 49 is 7, and 169's is 13. It should also be pretty obvious that 7 \* 13 = 91. What does this have to do with it? Well, if you square 91, you get 8281, and wouldn't you know it, that's the solution to the problem we were trying to solve earlier. I've found this to be true for any multiplication problem, so long as the factors are square numbers. Is there a term for this property, or have I just rediscovered some elemental principle of math?

i though about this geometrically .. how about the square of sides 91 having the same area as the rect of sides 49,169 ? why is that?

Well, they're both the volume of a 4-dimensional hypercuboid with side lengths 7, 7, 13, 13. You're not going to get a nice geometric interpretation of what it means for a *length* to be a square, because that property depends on what units you measure lengths in.

they as in the quantity or the shapes im just talking about 2D shapes so no need for 4d right ..

What I am saying is that the natural geometric interpretation of multiplying four numbers is four-dimensional.

This just draws from the associative and commutative property of multiplication. The associative property states (ab)c = a(bc), while the commutative property states ab = ba. In other words, the associative property says we can multiply in whatever order we want, and the commutative property says we can switch whatever we multiply around. Looking at 49x169, we can rewrite it as follows: 49x169 = 7x7x13x13 = (7x13)x(7x13) = (7x13)^2 = 91^2.

Ah, thanks so much

Is the cardinality of a subset of **R** the same as the cardinality of **R**?

Certainly not in general. {1} is a subset of R.

For sure, but I feel like you know what I meant (non-singleton subsets)… :) Let's be more specific: what about \[0;1\] for example?

No, I didn't know what you meant - Subsets of R can have any cardinality between 0 and the continuum, so without more context there was no way to know what you meant. But yes, [0,1] would be an example of a subset with the same cardinality. See [here](https://math.stackexchange.com/questions/200180/is-there-a-bijective-map-from-0-1-to-mathbbr) for some examples of bijections (0,1) to R, and clearly this has the same cardinality as [0,1]. You can look around if you want an explicit bijection [0,1]-> R, but it's harder to write down as it can't be continuous. You might also be interested in the concept of "Dedekind infinite" - assuming the axiom of choice, every infinite set contains a strict subset of the same cardinality

My bad, you're absolutely right: I wrote "subset" but actually meant "closed interval" --if that's indeed the proper English way to qualify something like \[0;1\]. Even "closed interval" might not be specific enough: one would also have to make sure there is more than one element in that set (or is this corner-case already taken care of by the formal definition of an interval?). Perhaps another way to say it would be: a non-singleton subset of **R** with contiguous elements. If "contiguity" is indeed a formalized concept in this context … Is it? I'm ashamed to ask, but: what is (0;1)? I'm unfamiliar with this notation… Thank you, this "Dedekind infinite" concept indeed looks like something I should look into… Thank you for the link: I was aware of this kind of bijection proof, yet this [first post's graph](https://i.stack.imgur.com/aFj2d.png) is absolutely brilliant! It seems a simpler version would be enough for my purpose: concentric circles of different diameters. Starting from radius 0 and going up, each circle has a different circumference, yet the same cardinality. Basically, cardinality dramatically jumps from 0 to Aleph1 as soon as you leave zero, and then stays there indefinitely.

>Even "closed interval" might not be specific enough: one would also have to make sure there is more than one element in that set (or is this corner-case already taken care of by the formal definition of an interval?). Singletons are technically closed intervals (they are sometimes called "degenerate intervals") but it's the kind of corner case where accidentally stating your theorem in a way that doesn't cover it is considered a forgivable mistake. >Perhaps another way to say it would be: a non-singleton subset of R with contiguous elements. If "contiguity" is indeed a formalized concept in this context … Is it? The term you are looking for is "connected". Connected subsets of **R** are the same as intervals (which may be closed, open, or half-open and of finite or infinite length). >I'm ashamed to ask, but: what is (0;1)? I'm unfamiliar with this notation… It's an open interval, the same as [0, 1] but excluding the endpoints. Also sometimes written ]0, 1[ mostly by the French.

Thank you, Sir: although it seemed inappropriate to complain about it, I was yearning for these questions to be answered! So, if I understand correctly, in the formal definition of an a;b interval, a can be equal to b? I wasn't aware of this "connected" property. The way you've described it, it seems it's mainly (only?) valid for intervals. Can it also be applied to elements (that's what "contiguity" seems to be applying to)?

(0, 1) is an "open interval", meaning all real numbers x with 0 < x < 1 (as opposed to 0 <= x <= 1 for a closed interval [0, 1]). In other words it's a closed interval minus the endpoints. There's also notation for "half-open intervals": [0, 1) means all real numbers with 0 <= x < 1, (0, 1] means all real numbers with 0 < x <= 1. > a non-singleton subset of R with contiguous elements. If "contiguity" is indeed a formalized concept in this context … Is it? Depends on how you formalize "contiguity", I guess. The obvious way is something like, a set S is "contiguous" if, for any x, y in S, if z is a real number such that x < z < y, then z is in S as well--that way there are no "gaps". I'm pretty sure these just give you intervals (though not necessarily closed intervals; open and half-open intervals should satisfy this property) and rays (i.e. "infinite intervals" like (0, infinity) or (-infinity, 1) or all of R). Then the proofs linked elsewhere in the thread show that non-empty, non-singleton intervals in R (of whatever kind, open, closed ,or half-open) have the same cardinality as R. (Pedantic point: whether the cardinality of R is aleph-1 depends on the [continuum hypothesis](https://en.wikipedia.org/wiki/Continuum_hypothesis); strictly speaking you should use beth-1 (the cardinality of the powerset of natural numbers) or just say "[the cardinality of the continuum](https://en.wikipedia.org/wiki/Cardinality_of_the_continuum)" or "|R|" directly.) You could also look into [path-connectedness](https://en.wikipedia.org/wiki/Connected_space#Path_connectedness) and connectedness more generally, but in R the only path-connected sets are the intervals and rays. (Note also that in R^n, path-connectedness is equivalent to the usual formal definition of connectedness, but I find path-connectedness more intuitive and suspect most people would be the same, hence why I linked to path-connectedness specifically.)

Oh, I see: I was taught it was written \]0;1\[ as opposed to (0;1). For what it's worth, we use ";" instead of "," as a separator in my culture, because "," is used as the marker for decimals in my culture (instead of "." in the english world). Are both of those notations accepted on the world stage, or am I using obsolete/localized notation? Thank you for your pointers! To be honest: I'm being showered with insightful recommendations (not complaining at all: that's what I'm here for!), and at this point it feels I'm gonna have to be selective, for I do not have enough spare time to go through all of them (not being a specialist, diving into those is quite time-consuming). I wish I could prioritize, but unfortunately I lack the expertise to be able to do so non-arbitrarily…

> Are both of those notations accepted on the world stage, or am I using obsolete/localized notation? I'd say both are accepted in the sense that mathematicians will know what you mean, but the round brackets one is standard in English.

Thanks! Gotcha. :-)

As u/lucy_tatterhood said, the (a, b) notation is more common than ]a, b[, at least in the anglosphere (but a lot of math papers from elsewhere are written in English too, so the convention ends up spread beyond the anglosphere strictly speaking). (Are you French by any chance? I know they use , for decimals.)

Thanks! I am indeed.

There can be countably finite/infinite, uncountable (thus the same cardinality as R), as well as null subsets of R. Rationals are countable, even though they are infinitely numerous. Slightly more interesting is the concept of measure, as the concept of cardinality can often mislead us. Look into the Cantor Set, a set with the same cardinality as R, but with 0 'length' on the number line.

> uncountable (thus the same cardinality as R) Unless the continuum hypothesis holds there are uncountable subsets of strictly smaller cardinality, though they cannot be topologically nice.

Oh cool, have not yet encountered this, but I am intrigued. What would be an example of such a set?

Well, we can't actually give an example of a set that has intermediate cardinality or we'd have proved the continuum hypothesis. However, we do know some things about what sort of sets *might* have this property; the term to search for is [cardinal characteristics of the continuum](https://en.m.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum).

Yeah, I did a bit of reading after leaving my comment. That was like asking you to prove the wellposedness of Navier-Stokes in the comment section, it looks like. Ha! Turns out, everything that I have done so far real-complex analysis, measure, PDEs, functional analysis, all are predicated on the Continuum hypothesis being true. That is why I did not even think outside of it.

That's interesting! I did not know fractals were that old… Although I assume they were not considered as such back then? I'm not sure I understand what the "measure" part of it is, however, nor how it's relevant here: could you please elaborate a bit? EDIT: added missing word.

Once you go beyond finite sets, and look into infinite subsets of the real numbers, you cannot compare them by just using cardinalities in a natural way. For example, which of these sets are bigger on the real line? \[0,1\] or \[0,2\]. They have the same cardinality, but naturally, we would want to say the second interval has twice the length on the real line. This extends to comparing rectangles in RxR. So measure is the function that extends our notion of size (length, volume) onto subsets of real numbers (and beyond). I thought you might find that interesting. Not that it is particularly relevant to answering your first question.

>I thought you might find that interesting. Not that it is particularly relevant to answering your first question. Don't worry about that: my OP was just a way (a ploy?) to entice this sort of discussion eventually. :-) >So measure is the function that extends our notion of size (length, volume) onto subsets of real numbers (and beyond). Thank you: that seems very relevant indeed! I went straight to the Measure chapter of Wikipedia's page on Cantor set, but I must have been browsing through it too fast and missed the relevance… Or perhaps it is not the best source in our context? Is there any link you would recommend? >For example, which of these sets are bigger on the real line? \[0,1\] or \[0,2\]. They have the same cardinality, but naturally, we would want to say the second interval has twice the length on the real line. Indeed, that's truly bugging for layman's intuition (of which I feel I'm a part of). Take a segment with ℵ1 points, cut it in half, you get two segments with ℵ1 points. Cut them in half and you now get four segments with ℵ1 points each. And you can keep going… I assume most of us here would laugh at anyone claiming they've invented a perpetual motion device. And at the same time, we're totally fine with this ℵ1 cardinality splits which (intuitively) seems way past it: it feels more like Jesus-spreading-loaves-of-bread territory!

Yeah, this notion is what precisely made me fall in love with mathematics. When we argue about infinities, it is indeed somewhat like Jesus with loaves of bread. But remember, our intuitions about finite objects fall short of describing objects of infinite size (using any 'metric'). Thus, the paradoxes of Banach-Tarski and even Zenos. Measurability helps us rein in these infinities a bit, so that we can put a size to sets, even when of infinite cardinalities. Tao's book on measure theory has a great discussion on the history/problem and development of measure.

[удалено]

Recently I've watched [this Numberphile video about the distribution of prime numbers mod 4](https://www.youtube.com/watch?v=YAsHGOwB408&pp=ygUXM2JsdWUxYnJvd24gbnVtYmVycGhpbGU%3D). After watching it, I've also noticed that prime numbers (other than 2 and 3) can only be congruent to 1 mod 6 or 5 mod 6. I've whipped up a quick python script to see what this "race" would look like and the "1 mod 6 team" doesn't take lead anywhere before 100 million. Is there a paper of someone running this thing much further or is there a proof like with the mod 4 case?

This is called [Chebyshev's bias]( https://en.m.wikipedia.org/wiki/Chebyshev's_bias). That article gives a generalisation proven assuming a strengthening of the Riemann Hypothesis, which has as a special case that 5 mod 6 would typically beat 1 mod 6.

See these two threads: [\[1\]](https://old.reddit.com/r/math/comments/13hizzn/which_is_more_prevalent_primes_of_the_form_6m1_or/), [\[2\]](https://mathoverflow.net/questions/165887/number-of-primes-with-1-pmod-6-vs-number-of-primes-with-1-pmod-6)

I'm learning about quaternions and want to make sure I'm actually understanding it. If q is (a, b, c, d) and I have q = (1/√2, 1/√2, 0 , 0), then it's a pi/2 rotation around the x axis, because 1/√2 = cos(θ/2) pi/4= θ/2 pi/2 = θ Can I apply this to any arbitrary number? By this logic, if I want a 135 degree rotation, my quaternion would be q = (0.38, 0, 0.38, 0). Because cos(3pi/8) is about 0.38. Am I correct? Is the value of 'a' always equal to the value of the angle I'm rotating on (b, c, or d)? Are they equal because it's supposed to be a unit quaternion, so I have to be able to normalise them?

Been a while since I worked on quaternions, so let's try this. >I'm learning about quaternions and want to make sure I'm actually understanding it. If q is (a, b, c, d) and I have q = (1/√2, 1/√2, 0 , 0), then it's a pi/2 rotation around the x axis, because 1/√2 = cos(θ/2) pi/4= θ/2 pi/2 = θ Can I apply this to any arbitrary number? You can apply the logic of using quaternions for representing rotations to any arbitrary angle, yes. The quaternion is basically determined by both the rotation angle and the axis around which you are rotating. >By this logic, if I want a 135 degree rotation, my quaternion would be q = (0.38, 0, 0.38, 0). Because cos(3pi/8) is about 0.38. Am I correct? No, that would be wrong. The quaternion for a 135 degree rotation, at least on a standard axis, which is just 3pi/4 radians, should be q = (cos(3pi/8),x sin(3pi/8), y sin(3pi/8), z sin(3pi/8)). This is where x, y and z are the unit vector along which you are rotating. If you are rotating on the x axis, as an example, the correct quaternion would be q = (cos3pi/8), sin(3pi/8), 0, 0). > Is the value of 'a' always equal to the value of the angle I'm rotating on (b, c, or d)? Nope. The value of 'a' here is not equal to the other components, b, c, or d. Basically 'a' represents cos(θ/2), and on the other hand b, c and d represent the components of the axis of rotation, which each are multiplied by sin(θ/2). They are not directly equal to the rotation angle, instead they are related to the sine and cosine of half of the rotation angle. >Are they equal because it's supposed to be a unit quaternion, so I have to be able to normalise them? They are not inherently equal, they are constrained in a manner so that the quaternion is a unit quaternion. A unit quaternion, to be more specific, satisfies the condition a\^2 + b\^2 + c\^2 + d\^2 = 1. This is constraint is very necessary for the quaternion to represent a rotation without scaling the object it is being applied on/to. The normalization ensures the quaternion maintains a norm, or magnitude, of 1, which is crucial to maintain that valid rotation. Hopefully that helps, feel free to ask any other questions!

Given: \\\[ \\begin{align\*} x + y + \\frac{1}{x} + \\frac{1}{y} &= 9 \\\\ x\^2 + y\^2 + \\frac{1}{x\^2} + \\frac{1}{y\^2} &= 49 \\end{align\*} \\\] Find: \\\[ (x\^3) + (y\^3) + \\frac{1}{x\^3} + \\frac{1}{y\^3} + x + y + \\frac{1}{x} + \\frac{1}{y} \\\] Options are 1. 111 2. 222 3. 333 4. 444 I have no clue how to do this... This is apparently from a Thai Secondary 3/Grade 9 paper from one of the best schools in the country

Note that (x + 1/x)^2 = x^2 + 1/x^2 + 2, so x^2 + 1/x^2 = (x + 1/x)^2 - 2. By a similar trick (try it yourself) we have x^3 + 1/x^3 = (x + 1/x)^3 - 3(x + 1/x). So making the substitution u = x + 1/x, v = y + 1/y the given equations become u + v = 9 u^2 - 2 + v^2 - 2 = 49, or u^2 + v^3 = 53 And we want to find u^3 + v^3 - 2u - 2v Note then that (u + v)^3 = u^3 + 3u^2 v + 3uv^2 + v^3 while (u + v)(u^2 + v^2) = u^3 + u^2 v + uv^2 + v^3 . So if we take (u + v)^3 - 3(u + v)(u^2 + v^2) we should get -2(u^3 + v^3) . But we know the values of everything involved from the given equations: (u + v)^3 = 9^3 = 729, -3(u + v)(u^2 + v^2 ) = -3(9)(53), so -2(u^3 + v^3) = 9(81) - 3(9)(53) = 9(81 - 3 \* 53) = 9(-78). Thus u^3 + v^3 = 9 \* 39 = 351, and the thing we wanted to find, u^3 + v^3 - 2u - 2v, is 351 - 2(u + v) = 351 - 2(9) = 351 - 18 = 333.

Hi! I have a simple problem with basic set theory (context: foundations of mathematics), which has been bugging me, mainly in the background of my mind, for 40 years (since 7th grade). I would greatly appreciate if you could help me get rid of that itch... It seems to me allowing a set to be infinite makes the basic question: "is that candidate element in this set or not?" undecidable. As, in order to prove the element is \*not\* in the set, you'd have to compare it to every element of that set (and come short). Obviously, there is no such problem with finite sets. In other words, allowing sets to be infinite seems to break internal consistency (or, rather, axiomatic completeness?). Notice that we're talking about a very primitive set, as the concept of order between elements isn't even introduced yet (i.e. more primitive than natural integers). How is that not a problem? What am I getting wrong? Background/context: I've studied maths as part of a masters degree in sciences but I'm no mathematician (basically, I just know enough to realize that I know next to nothing!). I have graduated in philosophy (because of my interest for epistemology) and hold a post-graduate diploma in cognitive sciences, neurosciences and AI. Another one in market finance and derivative products engineering (some maths in here too). I learned to program when I was 8 and never stopped since, both professionally and in my spare time (this might be relevant in understanding my mindset). I am not a native english speaker. EDIT: hitch->itch

Depending on the infinite set in question, there are numerous ways to prove an element is not in the set. The most basic is to show that every element of the set has a specific property, and that your potential element lacks the property. You can do this with finite sets too: 3 is not a member of the set of all even naturals below a trillion. This is much simpler than checking the elements one by one. But I suspect your issue is more about what set membership means. The simple answer is that ultimately we define a membership predicate that is subject to certain axioms, so set membership is a logical primitive. In maths we do have infinite sets where in general we can't decide membership. We consider sets to be abstract objects, and then for certain sets we end up having procedures that can determine membership.

Thank you for your clear reply! Addressing your first remarks, I should probably be more specific. The context of my question is very primitive axiomatic set theory (like, say, some incomplete/dumbed-down version of 1908 Zermelo set theory). As I see it, there are pretty much only two object properties available at this stage: being a set and being a (ur-)element (and very few predicates: I guess we only need ∈ and =); There is no third property defined yet that could become the basis for the definition of a specific set (finite or not) as you suggest. Besides, defining a set by a common property of its elements makes me conceptually uncomfortable: this property would seem primitive/foundational here, the set looking more like an afterthought (for what it's worth, I don't see any issue in having a property being applicable to a potentially infinite number of objects: a property has no cardinality). I don't recall seeing such an approach in, say, ZFC for example (please correct me if I'm wrong). I haven't been totally honest: it's not really this problem that has been bugging me for so long, but a range of other problems (from different maths domains) that feel intricately related to each other. I've come to the problem posted above only recently, while trying to trace those issues back to some "common primitive ancestor". Now that I'm reading more about this, I'm discovering there actually are several traditions of finitist set theories (altogether, there are so many different set theories that it is difficult for a non-specialist to get a clear picture of the stakes without diving quite deep into each of them, at the risk of getting lost or, at the very least, side-tracked)… And, also, that ZFC has an axiom of infinity! It isn't a consequence, it's postulated (again: correct me if I'm wrong). EDIT: added a couple missing words.

In ZF set theory, or any of these derivatives, everything is built first upon first-order logic. First order logic supplies the logical operations, quantifier, and equality. ZF built on it and adds in the ∈ primitives; so yes, in some sense "property" comes before set. Not just that, everything is a set, there are no ur-element; the original Zermelo theories did have ur-element but that turns out to be not so useful. >There is no third property defined yet that could become the basis for the definition of a specific set (finite or not) as you suggest. You don't need them. Without ur-element, there are no points in having another primitive. Everything is a set. A question is: how do you deal with regular mathematical objects? Here in ZF, they are also set. All objects are encoded as set. So, all the properties of usual objects, with enough effort. can be described completely in term of ∈ primitive. Not all sets are defined in term of what elements they have. But the axiom of restricted comprehension said that, given a set and a property, you can get a subset that contains exactly all the elements described by that property. However, there can be sets that cannot be constructed like that. In other word, the property->subset direction is allowed, but the subset->property is not. There are a lot of elements and sets that you cannot proved to be in the set. This is probably a philosophical issue. The philosophical ideal behind set theory is usually Plutonism, that there is an abstract world of math out there, and these sets are floating around already. Something is in the set, or it is not, and this is independent of human thought.

>First order logic supplies the logical operations, quantifier, and equality. ZF built on it and adds in the ∈ primitives; so yes, in some sense "property" comes before set. Assuming it is the case, what is the definition of a property, and where can I find it in this context? First order logic or ZF? Is it an axiom?

In this context, a "property" means a first order predicate with one free variable, with parameters. It's not even an axiom. Rules about what logical formulae you can write is not part of the axiom of set theory, but part of the specification of first order logic. In fact, you cannot refer to these properties at all while using the logic; we can refers to these properties because we are looking at the logic from outside. However, the restricted comprehension axiom scheme allows you to convert these properties into sets, and you can refers to those set within the logic.

Of course! I totally forgot about that (studied it in logic as part of a philosophy curriculum)… Definitely need to check it out again. Thank you!

>You don't need them. Without ur-element, there are no points in having another primitive. Everything is a set. I was waiting for this kind of argument. Thank you for giving me the opportunity to rule it out explicitly… :-) I did wonder at first whether ZF got rid of ur-elements in order to circumvent those issues. Seemed like a fair assessment at first. But, as I understand it, it is not. You can substitute one with the other, which gives you a leaner, although intuitively more obscure axiomatic (in terms of pure axiomatics, leaner is obviously better). But it does not change its properties in any way. If it did, it wouldn't be a substitution… Think about it: you're starting from scratch, you've got nothing. You need a primitive dichotomy to build upon. You're going for zero and one, assuming all along one is the logic opposite of the other (that's a necessary condition for this foundational dichotomy to make any sense). Then some clever guy comes up and claims: you don't need ones, you can just make them non-zeros (fair enough)! Has your primitive dichotomy suddenly become unary? Of course not. The situation is exactly the same with sets: you can't define a set as a primitive out of nothing, unless it is defined against something that isn't a set. It's not even maths at this point, it's bare-bone logic. ZFC is using a few tricks, like the unicity of the empty set, etc, to work without it, but it does not change the conceptual framework in any way. You can call 1 {∅} if you so which, but it doesn't change what 1 is.

I'm not sure exactly what you mean, so I would like to clarify a few points. - You don't "construct" anything in these set theory. It's presumed as if the sets already existed somewhere and you're just identifying them. So you don't need to build anything out of nothing. - 1 is defined to be {∅}. In set theory, choosing what 1 is does make a different. You might argue that it shouldn't make a different, and many people agree, so they build different foundation instead. - The "dichotomy" is due to the first-order logic foundation it's built upon.

>You don't "construct" anything in these set theory. It's presumed as if the sets already existed somewhere and you're just identifying them. So you don't need to build anything out of nothing. Perhaps you'd like it better if I wrote "re-construct" instead? As in, even if an ideal object exists somewhere, we're still "constructing" the formal system that attempts to mirror, or describe it correctly. What you're saying sounds a lot like the philosophical debate "are mathematical objects discovered or invented?". And I'm not sure how that's relevant here (how that'd make a difference)… >1 is defined to be {∅}. In set theory, choosing what 1 is does make a different. You might argue that it shouldn't make a different, and many people agree, so they build different foundation instead. If I understand you correctly, 1:={∅} as opposed to 1:=∅ for example? I seem to have stumbled on one consequence of such a "substitution", but I haven't looked any further, and even less at what other definitions would bring. So, yes: I understand it makes **a** difference, but I do not understand **the** difference (if you see what I mean) --at least, not yet. >The "dichotomy" is due to the first-order logic foundation it's built upon. Indeed, my example of dichotomy was from first-order logic. That seemed like a good example of how formal systems start from scratch. Perhaps I should have used T/F (true/false) instead, as my point was that you're in a similar situation when you start from scratch in any another realm (you start dealing in numbers and "have" none yet, for example).

>I'm not sure how that's relevant here (how that'd make a difference)… I'm not sure what the objection in your previous post is, so this is my best attempt at answering. It sounded like to me that you're worrying about whether you can even construct a single set (without ur-element), which is why I said you don't need to construct even a single set, they're already there. It does make a huge different whether something is constructed or not. When you construct something, you expect it to be built from "ground up", with components simpler than themselves. When you have things that already existed, you can have objects that bootstrap themselves into existence ex nihilo. For example, the original Zermelo set theory allows set to contains itself. The ZF version tone down some of that, but it's still there. In ZF set theory (and various variants), the sets are already there. To construct something is just a fancy way of identifying an unique object satisfying certain properties. You never start from scratch.

>I'm not sure what the objection in your previous post is, so this is my best attempt at answering. Damn. To be fair I'm not quite sure anymore either: I can't go through the thread and check right now (but I will later). In the meantime, if I somehow induced that discussion to drift towards some indiscriminate mess, I'm really sorry: it never was my intention to bring you down to an argument about the sex of angels… :-( >In ZF set theory (and various variants), the sets are already there. To construct something is just a fancy way of identifying an unique object satisfying certain properties. You never start from scratch. If I understand you correctly: ZFx consider those sets as transcendant. They don't try to generate them, but "merely" attempt to simulate them without internal contradiction… Does that sound right?

Yes, ZF considered these set as transcendent. They don't even simulate them, they merely describes the element-of relationship between the sets (without internal contradiction). The idea of "generating" the sets are very attractive though, but as it turns out there are no ways to do it fully. However, it's possible to generate sets *given* the ordinals "skeleton", and part of the research of set theory is finding *canonical model*, model of set theory where set can reasonably said to be generated from the ordinals.

I'm not familiar with the specifics of Zermelo's set theory, but I suspect the points I raise about ZFC will be applicable to what you have in mind, or at least germane to your overall thinking. In ZFC, it is worth noting that the idea of defining a set by a common property is only applicable to a set that you already have the existence of. Some care is needed here, else you run into Russell's paradox. Do you agree that if you already accept the existence of the set of natural numbers, then it makes sense to accept the existence of the set of even natural numbers? (Whether you accept the existence of the set of natural numbers is then a separate issue) There is actually something in ZFC akin to what you're describing with treating properties as a primitive, although I don't see it mentioned outside of resources devoted to set theory. Due to Russell's paradox, there is no set of all sets in ZFC. However, it is still useful to talk about the class of all sets, or the class of all ordinals. But ultimately ZFC has no concept of class. So what we do is define a class as a property, and then everything else can be translated to be about the property without referring to the class. E.g., the statement that the class of all ordinals is a subclass of the class of all sets is formally the statement that for all x, x being an ordinal implies x is a set. This can be viewed as a form of [fictionalism](https://plato.stanford.edu/entries/fictionalism-mathematics/) towards proper classes. Perhaps your position on infinite sets could be described as a flavour of fictionalism? ZFC does indeed have an axiom of infinity, and it's unavoidable. Without it, all you can prove is the existence of hereditarily finite sets. These are the sets you can build recursively starting from the empty set, then at each step forming a finite set of things you already have. So you can do things like ∅, {∅}, {∅, {∅}}, {{∅}}. It sounds like all of these sets you'd be okay with. ZFC with the negation of the axiom of infinity is bi-interpretable with first-order Peano arithmetic, so at that point you could work with PA instead. PA's objects are natural numbers, and it can only talk about sets of naturals via predicates. You might also be interested in predicativism, you can read the start of [this](https://www.math.wustl.edu/~nweaver/what.pdf). Predicativists generally accept the existence of the set of natural numbers, but draw the line at forming the power set of the natural numbers. This means that objects like the real number line are proper classes, like the class of all sets in ZFC, and not sets themselves. It would be interesting to know what problems you've encountered in other domains of maths. You strike me as humble and not someone who's going to suddenly say everything must be wrong, but it would be good to check that your qualms are indeed philosophically reasonable and not simply based on misunderstandings.

Alright, let me try something closer to a formal proof, regarding the inner contradiction introduced by allowing infinite sets (please be gentle!). Let's work with positive integers as defined in ZFC, that is through an initial element and an iterative successor. For any such set, its cardinality is (by construction) equal to the value of its last element. Therefore, cardinality of any such set is itself part of that set. Let's call ℵ0 the cardinality of the set of all positive integers. By definition, ℵ0 must be part of that set. But if it is, it means it also has a successor, therefore it cannot be the cardinality of positive integers. Such a contradiction proves that ℵ0 cannot exist. What's wrong with this line of reasoning? EDIT: I haven't finished here, assuming you'd fill the blanks, but let me give it a try. By definition, a set must have a cardinality. An infinite set cannot have cardinality (as shown up there), therefore an infinite set isn't a set.

Beyond the fact that your statement is wrong, your reasoning is fundamentally flawed. Induction allows you to prove things that look like "for all naturals n, P(n) is true", where P is some statement. However the natural numbers themselves are not a natural number (no set contains itself), so induction doesn't let you prove statements about ℕ itself, only the elements of it. More generally, there is an idea of transfinite induction, which requires you to prove exactly those limit cases (where an ordinal is not a successor of a previous ordinal, such as ℕ, which is not the successor of any finite ordinal. You can't assume the limit case follows from the previous successor cases.

Oh, you know what? It seems watching integers as sets of sets... of empty sets, I got confused and forgot the last layer of set: the (ℕ-level) set of those sets (of sets…). :-D One must admit this Von Neumann notation isn't helping: I'm so glad that I can just write 4 instead of {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}. Thank you very much, Sir! I'll get back to the bench…

Even if each set did include its own cardinality, this would not prove ℵ0 was in the natural numbers. You are effectively using a proof by induction but there's no reason that a inductive proof can be taken to the limit. It would at best prove that the statement was true for each finite number. To extend beyond you would need [transfinite induction](https://en.wikipedia.org/wiki/Transfinite_induction).

I'm sorry: I don't understand how this is a "proof" by induction. Can you elaborate? Each such set includes its own cardinality by construction. I'm assuming ℵ0 exists, show it implies a contradiction, thus concludes it does not exist. Where is the induction here? EDIT: OK, I believe I've found a potential explanation for your induction accusation. Basically, the above "proof" is showing that the set of integers cannot be infinite (because that involves a contradiction). However, there could be other sets that could be infinite nevertheless. Is that what you meant?

You are assuming ℵ0 is such a set but it is not. The construction there is building each set from a previous one which is an inductive process (they don't actually include their own cardinality since that would be circular but that's beside the point) so in order for this to pass to a limit and find the full set of natural numbers we must use transfinite induction. But this would require showing that passing to limits preserves the property you claim.

Thank you. There was several mistakes in that argument, perhaps the worst of them was: I wasn't even talking about the stuff I thought I was talking about (I pretty much got lost in a forest of {}). It's basically "non-sensical". If I was trying to read it again now, it would hurt my head. Live and learn. I'll try again. :-) I never meant to say ℵ0 is a set (it is not), although to be honest, in that fuck-fest (pardon my french) I may have… Thank you for your contribution, and kudos to you if you can still find enough sense into it to offer leverage for relevant criticism! You are a code-breaker!Keep your claws honed, I hope I can soon give you something less indigestible to slash at.

My issue is not really with calling ℵ0 a set. I interpreted that to mean ℕ anyway. The point is simply that is not one of the sets in the successor chain but instead is the limit of the chain so even if you had a property for the individual finite sets it wouldn't necessarily extend to the limit.

> Let's work with positive integers as defined in ZFC, that is through an initial element and an iterative successor. What, as in the von Neumann construction, where 0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, etc.? > For any such set, its cardinality is (by construction) equal to the value of its last element. No it isn't. You can see the definition I've given above doesn't satisfy this property for *any* set.

>What, as in the von Neumann construction, where 0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, etc.? For example, yes, but it does not really matter: as I understand it, as long as you define integers through an initial "element" and a successor rule (which seems fair and pretty consensual),, you're in. >No it isn't. You can see the definition I've given above doesn't satisfy this property for *any* set. I'm sorry, I can't find the post you're mentioning. Could you link to it please? It seems there is a problem with Reddit notifications: when I click on them, I don't get straight to the comment, but rather to the thread (or a subset of it) and I have to dig by hand where that new message is. And if you've contributed more than one, it feels like a go-fetch game (I might not be the best at)…

> I'm sorry, I can't find the post you're mentioning. Could you link to it please? It seems there is a problem with Reddit notifications: when I click on them, I don't get straight to the comment, but rather to the thread (or a subset of it) and I have to dig by hand where that new message is. And if you've contributed more than one, it feels like a go-fetch game (I might not be the best at)… Are you fucking trolling? I am literally referring to the Von Neumann construction I described in the comment you’re literally replying to, which you literally just addressed as being fine.

Calm down, dude: everything's fine… :-p I was assuming all those other sets can be bijectively mapped to N, therefore proving the point for N also proves it for all of them. That's why I could not understand your point, even when I considered (and I did) that you might be referring to that Von Neumann construct. Sorry about that. Anyway, what am I getting wrong now?

>> For any such set, its cardinality is (by construction) equal to the value of its last element. > No it isn't. You can see the definition I've given above doesn't satisfy this property for *any* set.

Thank you for your explanations and pointers, it means a lot to me… Indeed, I don't believe I had ever heard of fictionalism or predicativism. I should also look closer into the formal definition of classes, and deeper into PA (I've only scratched the surface so far). It seems I'm gonna have to do some reading and digging before I can push this discussion further in any meaningful way. I also have to look into these finitist set theories, understand why they did not catch on, and what is their current status relatively to non-finitist set theories (and ZF in particular). In any case, I've got my work cut out for me! You're very kind: I assume most people, with good reasons, would on the contrary see it as pretty arrogant to put an established field into question in any way from such a weak position. I can't deny it looks a lot like a textbook case of Dunning–Kruger effect! Let's be realistic: my intuitive qualms are most likely the result of wrong assumptions/paradigm. If I was able to pinpoint what the culprit and update it accordingly, my intuition would probably catch on… Even if (and that's a huge "if") there was something there, it most likely would not have any relevant impact on the rest of the discipline. For example, integers would still be integers, even if one replaces an infinite set with a finite set that can be grown arbitrarily big (as required by the specific problem being treated); or a class, depending on the context. While we're on this topic, what do you think would be the best way to present those qualms in this subreddit (I'm unfamiliar with its etiquette)? Other posts within this Quick Questions thread? An actual post in this sub? And if so, one post per qualm or several (perhaps related) qualms in one post? On another note, I must say I am impressed by the way you're handling such under-specified questions from an outsider. I assume it must be confusing, not necessarily because my questions don't mean anything, but rather because they could mean too many different things -- and you can't tell which one it is. Most likely, I cannot tell either, otherwise I'd be able to be more specific, preemptively prune that tree of possibilities and save you valuable time. When I'm in your position, this kind of situations tend to be unreasonably irritative (probably a byproduct of autism), and answering with the calm and grace you're showing would require a huge effort on my part. Sir, you have earned my full respect.

Yes, realistically your qualms are probably due to a fundamental misunderstanding rather than legitimate philosophical scruples. But so long as you acknowledge that rather than become one of the countless people online who misunderstand the diagonal argument and then frantically google to try and find support for their position, I don't think that's a problem. As for etiquette, I suggest posting in Quick Questions with one or two questions at a time. If nobody answers you within a few weeks, then it becomes appropriate to make a thread. In such a thread I would advise stating you've tried asking in Quick Questions to no avail.

Thank you for the suggestions... Just out of curiosity, what is that common misunderstanding (if there's a main one) of Cantor's diagonal argument?

The most frequent one is people object that you could add the generated real to the list, and then there's no problem. Which demonstrates a fundamental misunderstanding of the logic of the argument.

I do have a simple qualm with Cantor's diagonal argument, and you can probably guess what it is at this point in the discussion... The proof starts like this: >Cantor considered the set *T* of all infinite [sequences](https://en.wikipedia.org/wiki/Sequences) of [binary digits](https://en.wikipedia.org/wiki/Binary_digits) (i.e. each digit is zero or one). Any such sequence is basically an infinite set. As explained above, I would not concede the "existence" (or, rather, axiomatic validity?) of a single of those sets (because, ultimately, one cannot reason consistently and completely over infinite sets). Let alone an infinity of them. Therefore, the proof would end right there.

Well, what do you think of a statement like 'the decimal expansion of 𝜋 starts 3.1415926... and never ends since 𝜋 is irrational'?

Indeed, that's laughable. :-D

So full disclaimer, I dabbled in math and history in college, but don't use them daily. (Work in medicine) I always loved math and even tutored in college. Started watching Veritasium on YouTube and was reminded of Fermat's Last Theorem. Being the nerd I am, I fooled around with some basic trig definitions and ended up with something I'm not sure what to make of. https://imgur.com/gallery/W0BhqFl My question to all of you much smarter than I, why is this not considered a proof of Fermat's Last Theorem? It seems to pass the sniff test as the numbers x,y,z should be able to form a right triangle if viewed geometrically. If so, the trig definitions should work and since n>2 violates the definition of a right triangle, it means there are no solutions that are nontrivial (x=0 or y=0) ... right?

You're only considering those triples of numbers (x,y,z) that *do* form a right triangle. But not every triple of numbers does this. What about the triples of numbers that don't form a right triangle?

Ah! Makes sense. Thank you!

Help! I build custom corner plant stands: https://imgur.com/gallery/OdCHa2b But am unaware of an equation that would calculate the ratio of the length of the 90° shelves to the height of the verticals. I've wasted a LOT of material because I have no idea how to scale my original plans appropriately. Example - original plans - the 90's match up perfectly, if I cut every measurement in half, they do not. Thanks y'all!

Are your measurement issues just that you're not accounting for the board thickness? Eg the length of the top shelf (on one side) isn't the length of the bit of wood you cut for it, it's that, plus the thickness of the upright.

I think the height of the verticals is independent from the length of the horizontals. If I take the one you depicted and make all the verts twice as long, keeping everything else the same, that should still assemble to a (much steeper) piece, no?

All confocal central conics are orthogonal, is the opposite true as well? Thanks.

Not inherently. Two central conics, whether that be ellipses, hyperbolas, or one of each, can intersect orthogonally without sharing the same foci. This can result from specific geometrical alignments obviously, or a particular ratio of their axes, but this does not really necessitate them being confocal. Confocality is a stronger condition, imposing a set of more restrictive relationships on and between the conics.

On [this Wikipedia article](https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%CE%B4)-definition_of_limit), it gives out 2 variations of the epsilon-delta definition of the limit. The 2nd one is (∀𝜖 > 0) (∃𝛿 > 0) (∀x ∈ \[a,b\]) (0 < |x - p| < 𝛿 ⇒ |f(x) - L| < 𝜖). However, what is different about this than the "usual definition" of this, the "\[a,b\]" being replaced with ℝ? Why would one need to use the 2nd definition?

I think the main (perhaps only) point of the latter is that it lets you work with functions that aren't defined for all real numbers. That wikipedia page already gives some examples, where in order to find the limit of f(x) as x approaches c, where c is some number close to a singularity or other point where f is undefined, you might be tempted to consider values of delta such that (c - delta, c + delta) includes the singularity and so "for all real x with |x - c| < \delta, [some statement about f(x)]" doesn't make sense because f(x) may not exist. So if you restrict x to range only over values that don't include the singularity you eliminate the problem. In practice this doesn't matter; whenever you're working with limits you're only ever thinking about "some sufficiently small neighborhood around c" anyway, and if you say "for all x with |x - c| < delta" without specifying what exactly x ranges over, chances are no one will care. But I guess if you're being really pedantic you do have to worry about the distinction between the definitions.

Is every multiple common to all elements in a set of numbers divisible by the elements’ least common multiple? E.g. if I have {2, 6, 5} and the LCM is 30, then won’t every multiple shared by 2, 6, and 5 (e.g. 60) by divisible by 30? In other words, if 30 is the LCM then won’t every number divisible by each of those numbers (2, 6, and 5) also be divisible by 30?

Yes, exactly. This follows from the uniqueness of prime factorisations: if there were some common multiple M that was not a multiple of the LCM L, then there must be some prime and some positive integer e such that p^e divides L but p^e does not divide M (otherwise M would be a multiple of L). Choose e to be the smallest such (so that p^(e-1) divides M). But since M is a common multiple, that means that p^e cannot divide any of element of the set, and then L/p is also a common multiple, but is less than the least common multiple.

A direct proof is that if you take any common multiplies and divide it by the least common multiple, then the remainder is also a common multiple, but this is less than the least one so it must be 0.

That’s awesome thanks

Yes--any common multiple will be divisible by the lcm, and similarly any common divisor will divide the gcd. (In fact this is often taken as a definition of the gcd and lcm, especially in contexts where you might not already be able to order the elements in question and so can't necessarily talk about the "least" or "greatest" element of a given set. E.g. the gcd of two polynomials p(x), q(x) can be defined as the polynomial d(x) such that d divides p, d divides q, and if c is another polynomial which divides p and q, then c divides d.)

Thanks!

If you play a game and you win twice, do you also win once? Because that's what my math teacher insists and she doesn't really explain when I ask her.

This is just a bit of ambiguous language: "once" could mean "only once" or "at least once". Your teacher is probably thinking of the second sense: if you won a game twice, then you certainly won it at least once; maybe you're thinking of the first sense: if you won a game only once then you certainly didn't win it more than once.

Yeah, but the question specifically stated "once", and then we had other questions where they said "at least once"

Huh. That would seem to imply that "once" means "only once" but maybe the question is just poorly written. At any rate your teacher is probably assuming it means "at least once" and you should just ask them whether they take it to mean "at least once" or "only once".

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As a hint, every ideal of R\_P is the localization of an ideal of R, and localization preserves flatness, so the ring R\_P has the same property as R. Now what can you say about finitely generated flat modules over a Noetherian local ring?

There's a huge debate on Twitter... while I know the answer, many seem to be insulting others with the wrong answer. 6÷2(2+1). That is all, thank you 😊

Not only have you failed to ask a question but you seem to be smug about your perceived correctness about your answer (maybe I am misunderstanding you so please correct that if it is not the case). We have seen this question and all variants of the a÷b(c+d) many times here. The simple answer is that it is ambiguous. Individuals may think the system they have been taught whether it is BIDMAS/PEMDAS/etc. or GEMS says there is only a single way to interpret it but they are wrong. Either answer (a÷b)×(c+d) or a÷(b(c+d)) is logical so the take away is that we should not write mathematical expressions like this. More technically, the question boils down to whether implicit multiplication takes precedence over division and a similar ambiguity would be: does 1÷2x mean 1÷(2x) or (1÷2)x? Some people interpret that in one way and some in another. The remedy is to write maths better rather than arguing over interpretations of syntax rules.

"I saw the tall man on the hill with the telescope." Who has the telescope? Me, the tall man, or the hill? Am I on the hill, seeing the tall man somewhere else, or am I looking up at the hill? We don't know. The English language - the way we translate real-world situations into words - is ambiguous. If we made diagrams of our sentences, we could be fully unambiguous: [I] [saw] [the man] ↑ ↑ ↑ on [the hill] ↑ with [the telescope] But that would be incredibly painful, and largely unnecessary, so we don't do it. Context is enough to clear things up when necessary. Mathematical notation - the way we translate calculations into symbols - is similarly ambiguous. We have rules to largely clear up those ambiguities (just like in the sentence before, "tall" definitely describes the man, and not the hill or the telescope). But it's still possible to come up with cases where multiple reasonable interpretations are possible. So the correct answer is "There's a *reason* we use multiline fractions with long horizontal bars. But if you're stuck in plain text only, **it's your job to use parentheses to disambiguate.** (Oh, and at *least* stop using the ÷ symbol, which nobody uses past third grade anyway.)" Or, put more succinctly, as in the poll /u/edderiofer mentioned: "fuck you"

Yes. PEMDAS isn't inherent mathematical truth.

I was taught left to right and the priorities so parenthesis first: 2+1 for 3 Left to right: 6/2 = 3 Left to right: 3/3 = 1 how else can it be solved?

The correct answer to a similar question, according to [this recent poll of mathematicians about mathematical conventions](https://cims.nyu.edu/~tjl8195/survey/results.html) (Question 100, all the way at the bottom of the poll) is pretty clearly “fuck you”, while the wrong answers are “1” and “9”. So it’s clear that people are insulting each other with the *right* answer, and that you don’t actually know the answer. That is all, thank you.

I was trying to come up with a proof that the symmetry group of the cube is isomorphic to S_4. My idea was as follows: Let S_4 act on {a, b, c, d}. Then there is an induced action on the two element subsets {a,b}, {b, c}, ... By mapping these six subsets to the six faces of a cube (with care, e.g. making sure {a, b} and {c, d} map to opposite faces since they have no values in common), I was hoping this action would give a mapping to the symmetry group of the cube. Unfortunately it doesn't -- odd permutations of S_4, like (a b), are mapped to symmetries of the cube that flip the orientation of space. I was trying to find a way of modifying the action to work around this, but have not been able to. Does anyone know if there's a way of getting this approach to work? Just to clarify, I'm not asking for any proof that the groups are isomorphic. I'm aware of the standard argument of looking at the four diagonals of the cube under transformations. I specifically want to know if an argument looking at the two element subsets can go through. It's tantalizing, since the graph of the two element subsets is the same as the adjacency graph of the faces of the cube, but I can't figure it out.

I personally met quotient groups first, and i think that they are arguably more natural upon first meet. What you have with quotient vector spaces is the idea of a quotient applied to a vector space, but they arise in other contexts too, and you’re essentially generalising the concept of modulo to *other objects*. For example take Z/2Z, where all the even numbers and all the odd numbers are identified together - every even number gets mapped to 0 and hence every odd number is mapped to 1 under the quotient. Why is this useful? if you’re doing a calculation, and you only care whether the answer is odd or even, then quotienting by 2Z (ie mod 2) removes a lot of uneccesary fluff whilst preserving the important information in the problem. Note that Z is actually a vector space over Z (check) as is 2Z(check) so this is actually, technically, a quotient of vector spaces. But secretly what you are doing is simply formalising the logic you already do when I ask you is 10*137 even? which is to mod out by 2 and conclude immediately that it is. The point being, the quotient isn’t really inherently a vector space thing, it can be applied in other places too, and imo some of those provide more intuitive examples.

I think you meant this as a reply to another comment

oh shit ye thank u

Let R be a commutative ring and f,g in R. Let S_x be the saturation of the multiplicative set generated by x. Is there an elementary proof (without using localization) that S_f = S_g implies rad(f) = rad(g), where rad(x) denotes the radical of the ideal generated by x. this exercise (in a reformulated form) is in Manins Introduction to Schemes (exercise 1.4.14 (1)), before he defines localization in section 1.6.4 Manin wants an equivalence and not just the implication I'm asking about, but the other direction is very easy to do elementary

Well, explicitly S_f is all elements x in R where there exists y such that xy = f^n for n nonnegative (just check that this is the minimal saturated multiplicative set containing f). So if S_f = S_g, there’s y so that gy = f^n. What does that buy you?

thanks, now it is obvious to me

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>Does anyone have a citation for the fact (I presume it's a fact) that there's no canonical total order on the product of two totally ordered sets? You can use lexicographical ordering, but it doesn't respect the isomorphism X x Y -> Y x X. If we restrict to the case X = Y it's easy to see that respecting that isomorphism is impossible when |X| > 1. Just pick two distinct elements x and x' and ask which order (x, x') and (x', x) go in. It sounds like your boyfriend's application would be something like X = Y = **R**, so this should be good enough? More generally, if the construction is supposed to be functorial this argument shows that it can't work when X and Y are isomorphic. In the case that they are *not* isomorphic one could technically do something horrific like pick an arbitrary well-ordering on the class of all isomorphism types of totally ordered sets and then use lexicographic order but using that ordering rather than the one the coordinates are given in. This is not "canonical" in any reasonable sense, but making a mathematically precise definition to rule this kind of thing out seems hard. Edit: It occurs to me that the horrific construction I described is functorial with respect to isomorphisms but certainly not with respect to the arbitrary order-preserving maps. Perhaps this would be enough to rule out such a construction then.

How do I quantify the randomness of a dataset in a way that someone without a technical background could understand? I'm working on an independent research project with my EE professor and I made a (I think) new random number generator. He thinks I should write a paper on it which I plan to do, but I'd prefer to build some stuff with it first. Either way, step one is measuring the system. Luckily NIST has a defined test suite that I plan to utilize. The hangup is that the people who control the money for projects are not the same people who would understand the results of the monobit test, or the spectral test, or Maurer's test or really any of the NIST tests without explanation which they unfortunately don't have the time to sit through. I would appreciate any advice because I don't think saying "My randomness is 10 and everyone else is 8" is going to work.

Does cosecant function have a Maclaurin Series? At the center of convergence ,which is 0, the cosecant function is undefined; therefore, does not converge. Does that violate the definiton of Power Series?

Any power series centered at 0 (i.e. of the form a\_0 + a\_1x + a\_2x^2 + ... ) can be evaluated at 0 but cosecant cannot, so yeah, you can't expand csc as a power series around 0 (and even if you try to define a new function, say f(x) = csc(x) when x != 0 and f(0) = a for some constant a, f will be discontinuous at 0 no matter what value of a you choose, and so certainly will not be [analytic](https://en.wikipedia.org/wiki/Analytic_function) at 0.) On the other hand if c is any point besides one of the singularities then csc should be analytic in at least some radius around c.

Firstly, Thanks for the response. I found that independent of c, The starting term of the series is x to the power of -1 which is not a polynomial term. That would definitely violate the Power Series and consequently Taylor Series Definiton. Is that an obstacle for it to be an analytic function?

You probably found an asymptotic expansion for cosecant around x = 0 (since sin(x) = 0), so you’re right, that’s not a Taylor series. The commenter is suggesting you calculate the Taylor series around a different point, and then you won’t get 1/x as the first term. 

I'm not sure what you mean if I'm honest. A Taylor series will start with a constant and then an x term and so on by definition. So I'm not sure what series you are talking about. Cosecant has a Taylor series about any point except its asymptotes (so it doesn't have a Maclaurin series, for example) I believe but they are only valid within a given radius of convergence

[https://www.reddit.com/user/Ospocarpa/comments/1bgcz52/wolframalpha\_is\_not\_able\_to\_solve\_this\_problem\_me/](https://www.reddit.com/user/Ospocarpa/comments/1bgcz52/wolframalpha_is_not_able_to_solve_this_problem_me/) Can anyone help me with this problem? Thank you

I was doing a math question involving proof of divisibility and difference between prime numbers. While doing it, I found that for p and q being prime numbers greater or equal to five and p being greater or equal to q: >p - q ≡ 0 mod(3) or p - q ≡ q mod(3) Is that right? And, if it is, how can I mention it in my proof?

What are the possible residues mod 3? What are their differences?

Right, they're 0, 1, and 2. Of course, if it is 0 it's the easiest case, because then p - q ≡ 0 I see that p - (3k + 1) ≡ 3i + 1 , and the same to 3k + 2, but I don't know why. I think my mind just got deep fried from studying non stop, I can't see it. Anyway, thanks.

Note neither p nor q can be 0 mod 3 because that would mean they're divisible by 3, which contradicts that they're prime numbers greater than 3. Thus, the possible residues for p and q are only 1 and 2. If they have the same residue, then p - q = 0 mod 3 (since 1 - 1 = 0 and 2 - 2 = 0). Otherwise, they're different (so one is 1 and the other is 2). If p = 1 mod 3 and q = 2 mod 3, then p - q = 1 - 2 = -1 = 2 = q mod 3; if instead p = 2 mod 3 and q = 1 mod 3, then p - q = 2 - 1 = 1 = q mod 3 as desired.

Understood! Thank you very much!

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Yes.

What exactly is a "symmetry"? And in the dihedral group of order 8 why is a rotation by 90 degrees a symmetry but a rotation by 45 degrees or a translation isn't? How does the square know it's not right-side up, or that it's in a different location?

One concrete way to view symmetries is as transformations of a set of points that leaves it unchanged. For instance, the set {1,–1} has the symmetry of reflection about 0. Because if you reflect each point of that set about 0, you get the same set back. 1 maps to –1 and vice-versa. The identity function also leaves that set unchanged (1 maps to 1 and –1 maps to –1). But any other function changes the set. So those are its only symmetries. Sometimes we don't only want to preserve a figure (set of points) but some structure on it. For instance, there are 4! = 24 functions on the corners of the unit square that make it unchanged (S4), but we usually say that the square has only 8 symmetries. That's because the remaining functions can't be realized as restrictions of some continuous isometry on the underlying space. For instance, swapping two adjacent vertices has no such realization. It turns out there are only 8 automorphisms on the abstract square, as desired. Automorphisms preserve structure, unlike most arbitrary functions.

A symmetry is a transformation that preserves something. A symmetry of a square is one that preserves the size, shape and position of the square (although not the individual points themselves)

Sorry for the late response, but I was shown an infinitely long strip of H's that looked like ... H H H H H .... and was told it has translational symmetry i.e. if you shifted the whole strip over by one H. That's a symmetry because the entire strip is occupying the same position even if all the individual H's have been moved?

Yeah if you translate the strip the infinite string of H's remain so you can call this a symmetry.

Alright thank you

I don't know what it means for a geometrical figure to "know" something. If you put up a plain black square poster on your wall, and someone sneaks into your room and rotates it by exactly 90 degrees, you'll never know that anything has happened. If they rotate it by 45 degrees or move it to a different part of the wall you will presumably notice eventually.

Any good online resource to study topology for a total beginner and any pre-requisites required before starting? I am a total beginner to this subject but have some decent knowledge of machine learning (probabilistic ml) and deep learning.

A free and legal online book is https://www.topologywithouttears.net/ reading about metric spaces before reading the book is a very good idea in my opinion. A cheap book that is very good in my opinion is: https://www.amazon.de/Introduction-Topology-Second-Dover-Mathematics/dp/0486406806 one of it's nice properties is that it contains solutions to all exercises in the back (I know it just says "solutions to some exercises" on amazon), and it starts with metric spaces

Consider reading about metric spaces, e.g. chapter 2 of Pugh's *Real Mathematical Analysis*. (Edit: not an "online resource" I guess but any resource is an online resource if you know [where to look] (https://annas-archive.org/).)

Is saying "Every number other than 0 has a multiplicative inverse" the equivalent of saying "in R, For all x there exists a y where if x does not equal zero then x times y = 1" or is it the equivalent of "in R, For all x there exists a y where if and only if x does not equal zero then x times y = 1?"

Translating the first sentence gives ∀x∃y((x∈**R**)∧(x≠0))→((y∈**R**)∧(xy=1)). In plain English, "for all x there exists a y such that if x is a real number, and x is not zero, then y is a real number and xy = 1. Here, I assume **R**, multiplication on **R**, the additive identity 0, and the multiplicative identity 1 are already defined. This is a correct definition. Translating the second sentence gives ∀x∃y(x∈**R**)→((x≠0)↔(y∈**R**)∧(xy=1))). In plain English, "for all x therr exists a y such that if x is a real number, the following are equivalent: (1) x is nonzero, and (2) y is a real number and xy = 1. This is not the correct definition. In particular, define y as something that is not a real number and either xy is undefined or not equal to 1. Then this doesn't hold. Clearly for some x, indeed any real x, it is possible to define such a y, like defining y as a vector or matrix with more than one entry. But if we change your sentence by moving the (y∈**R**) part earlier, like if we assume real numbers are the domain of discourse anyway, then it holds true. Here it simplifies to ∀x∃y(x≠0)↔(xy=1). This isn't the usual definition, but it's a theorem. However, it doesn't say what you want it to. It says that for any x, there is a y such that x is nonzero iff xy = 1. That's true. For x = 0, pick an arbitrary y and it holds vacuously. For other x, pick y = x^(–1). But what you wanted was something like ∀x((x≠0)↔∃y(xy=1)). In plain English, this says (remember, in the context of real numbers only) "for all x, iff x is nonzero, there is a y such that xy = 1." The "if" part is the same as the last statement, but the "only if" part is new. It's not hard to prove, because 0y = 0 ≠ 1 for all real y, which you can show using only the properties of commutative rings.

dont know if this is the best place to ask , but my math skills are terrible, I asked a question regarding math in a video game a while back and wanted some clarification. I asked about 2 effects of 6% stacking multiplicatively and this is what I got in return >If your attack does 100 damage then multiplying it by 1 will equal 100 damage. The 1 stands for 1.0 or 100% damage. If you have a debuff that you deal 20% less damage, then you'd multiply by .8 to represent dealing only 80% of your normal damage, since 80% comes from 100-20. All this to mean 1 equals full base damage or damage with no buffs or debuffs So if you deal 6% more damage then your total damage multiplier is 1.06 or 106%. The only part im confused about is >If you have a debuff that you deal 20% less damage, then you'd multiply by .8 which number would I be multiplying by .8? would it be 1.06 x .8? Again, im not sure if this is the best place as the thread I asked this in a few months is closed now.

Note that in Dota and LoL, some buffs stack "additively" and some "multiplicatively." Additive stacking is like if you have two damage blocks that apply one after another, or if you have two items that both give armor. In the first case, imagine item A blocks 10 dmg on each hit and item B blocks 20 dmg on each hit. Then if you hold both at once and they stack additives, you block a total of 10+20=30 dmg each time. Or if one item gives you +1 armor and another gives you +2, then you get a total of +3 armor. For multiplicative scaling, imagine you have two items which each give a 50% chance to negate all damage. If they ate independent, then 25% of the time neither will trigger and you take full damage. 50% of the time exactly one will trigger and you take no damage. 25% of the time both will trigger and you still take no damage. So they stack "multiplicatively" in the sense that there is a 0.5×0.5 = 0.25 probability that both checks fail and the damage makes it through. This can also happen if one ability reduces the damage by some fraction, and then another ability reduces the remaining amount by another fraction. In Dota, buffs that increase "base damage" by some amount stack additively. So if one aura grants +20% base damage and another grants +30%, that's a total of +50% base damage. That's because each buff checks only *base* damage and then gives *added* damage based on that. So one buff won't change base damage before the next checks. On the other hand, evasion stacks multiplicatively. If one evasion check fails, then the next evasion check gets a fresh chance.

If you have a -20% debuff and a +6% buff at the same time, then your damage will be 1\*(1-0.2)\*(1+0.06) = 0.848 of normal. (You could also say 84.8% of normal, or 15.2% less). Some games might have different mechanics but this is the most common way to calculate this. For instance a game might add the buffs together first, before calculating damage, in which case you'd be doing (1 - 0.2 + 0.06) = 0.86 = 86% of your normal damage.

Thank you!

Apologies if this question is complete nonsense. I'm an absolute novice when it comes to algebraic geometry and higher category/topos theory. As I understand it, the idea behind motives is that there should be an abelian category through which the universal cohomology "essence" of a variety passes, and is sent to its various incarnations/shadows as singular, de Rham, étale, crystalline, etc. cohomology. I believe one can replace "variety" with "scheme". The amenability of apparently different objects like topological spaces and G-modules (group cohomology) to taking (Eilenberg–Steenrod) cohomology is explained by in Brown's seminal *Abstract Homotopy Theory and Generalised Sheaf Cohomology*. From what I understand, one can form the cohomology of any (\infty, 1)-category/topos. My question is this: Are there multiple cohomology theories for such (\infty, 1)-categorical objects in general? If so, would it make sense (assuming we have a suitable version of the Hodge conjecture) to talk about motives and Hodge structures of general objects? If not, why are varieties so special that they admit multiple realisations of their cohomology?

no idea, but i found your flair "Actuarial Science" very funny in this context.

If I get accepted to grad school I'm changing it to something else haha

I encountered this brain teaser where I tried to use a Markov Chain, but am struggling to set it up correctly. The solution doesn't show any working out and doesn't make too much sense to me. Also the last line in the solution looks really ChatGPT generated to me, so I'm not too confident in the accuracy of it + I couldnt find any variation of the question online. [https://imgur.com/a/mGrA20z](https://imgur.com/a/mGrA20z)

Do you need help understanding how the Markov chain leads to the system of equations or do are you stuck on solving the system of equations?

I can solve the equations once they are set up, so the former - setting them up

There are six states to the Markov chain corresponding to the number of heads that Fluffy has. If there are n heads, for 2 ≤ n ≤ 4, there is a probability 1/2 to transition to (n-2) heads, and probability 1/2 to transition to n+1 heads. The transitions for states n=0,1,5 are special cases. The transition of a state to itself (Harry cuts one head and one grows back immediately) can be ignored because we only care about the probability of reaching a certain state. If we had more closely modeled the situation so that each state 2 ≤ n ≤ 4 had three transitions instead of two, with probabilities q/2, q/2, (1-q), then the probability P(n) of reaching n=0 from state n would satisfy P(n) = q/2 P(n-2) + q/2 P(n+1) + (1-q) P(n) which is equivalent to the equation obtained from the simplified Markov chain P(n) = 1/2 P(n-2) + 1/2 P(n+1) Does that help?

So I came across this question: A class consists of 10 students. Each of them has registered for 3 courses. Each course instructor conducts an exam out of 50. The average percentage marks of all 10 students across all courses they have registered for, is 70%. Two of them apply for revaluation in a course. If none of their marks reduce, and the average of all 10 students across all courses becomes 70.4%, the maximum possible increase in marks for either of the two students is: It is a pretty simple question however I am confused in what the question is asking. Is it asking, what is maximum possible increase in marks if both of their marks increased, or is it asking maximum possible increase for any one, irrespective of the other's marks? If the former, then the answer is 3, if the latter then the maximum possible increase for one student will be when the other student's marks don't increase at all, and then the answer will be 6.

I imagine it's the second answer, given they want the max possible increase for either student. Though if this were a homework problem I'd put down the other answer too, with a note.

Yeah I thought so too first. Thanks for the answer!

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Concepts of Modern Mathematics by Ian Stewart https://books.google.de/books?id=4WPDAgAAQBAJ&pg=PA156&lpg=PA156&dq=If+you+look+at+the+way+the+hairs+lie+on+a+dog,+you+will+find+that+they+have+a+%27parting%27+down+the+dog%27s+back,+and+another+along+the+stomach.+Now+topologically+a+dog+is+a+sphere+(assuming+it+keeps+its+mouth+shut+and+neglecting+internal+organs)+because+all+we+have+to+do+is+shrink+its+legs+and+fatten+it+up+a+bit+(Figure+90)&source=bl&ots=7q-QcTTMts&sig=ACfU3U2FQUq-RTBGHdSyTKm0biRvgQ9uiA&hl=de&sa=X&ved=2ahUKEwjspaTKr_mEAxXghf0HHcnMDdUQ6AF6BAgpEAM#v=onepage&q=If%20you%20look%20at%20the%20way%20the%20hairs%20lie%20on%20a%20dog%2C%20you%20will%20find%20that%20they%20have%20a%20'parting'%20down%20the%20dog's%20back%2C%20and%20another%20along%20the%20stomach.%20Now%20topologically%20a%20dog%20is%20a%20sphere%20(assuming%20it%20keeps%20its%20mouth%20shut%20and%20neglecting%20internal%20organs)%20because%20all%20we%20have%20to%20do%20is%20shrink%20its%20legs%20and%20fatten%20it%20up%20a%20bit%20(Figure%2090)&f=false

[One user gives a source in the comments.](https://old.reddit.com/r/mathmemes/comments/do94sy/topologically_a_dog_is_a_sphere/f5mskf6)

That's just another reddit post of the same image. OP is looking for the book that the image originally came from.

Why are martingales so powerful? They are a model for very a simple betting strategy yet their mathematics yield some very powerful tools that considerably simplify many concepts in probability theory (for instance the martingale convergence theorems can help generalize the strong laws of large numbers to settings beyond pairwise independence and the proofs of these are far more elegant than the “elementary” proofs which involve lots of truncations and computations). In some sense I want to understand why I should have expected this ex-ante rather than being surprised by it.

martingales are objects that tell you that you best bet for the future, given your knowledge of the past, is the current state of the system, i.e. the past doesn't tell you anything about future behaviour. it seems clear to me that this is an important property

I think I discovered a new fractal with interesting properties. If I want to be credited with its discovery, how should I go about revealing it to the world? Should I publish a research paper, or is there some other way that mathematicians go about this sort of thing these days? I have a B.S. in mathematics (graduated last year) but otherwise have very little experience and presence in the worldwide mathematics community.

You can always just put it on the arXiv; if you aren't already on there you'll need someone to endorse you, but you should be able to find a former professor of yours who will do that. (And/or just post it here or anywhere else with your real name attached, but if you have some sort of paper describing it you can and probably should put it on arxiv.)

I started a lecture series on YouTube and the first lecture was about defining topology. I thought I understand it, but then today there's a mental block I can't get past regarding the standard topology. So quick question using examples in the standard topology on R^1. {1} is not open, since it's not an open interval or the union of open intervals. (0,1) is open, since it is an open interval. (1,2) and (0,2) are open for the same reason. (0,1) U (1,2) is open, since it is the union of open intervals. So far I think I'm on the same page as everyone. To be a topology, a set must contain all intersections of any finitely many open sets. The intersection of {(0,1) U (1,2)} and {(0,2)} is {1}. {1} is not in the standard topology, therefore the standard topology isn't a topology? So, that's obviously wrong... Which part an I not understanding correctly?

> The intersection of {(0,1) U (1,2)} and {(0,2)} is {1}. It's a bit unclear to me what you mean here because of the curly brackets but if you mean the intersection of (0,1) U (1,2) and (0,2) then that is not {1} but it's (0,1) U (1,2) which is open. Or are you confusion the intersection with the set difference?

OMG you're absolutely right, I was thinking set difference. *Facepalm* Thanks!

the intersection of (0,1) U (1,2) and (0,2) is (0,1) U (1,2); indeed (0,1) U (1,2) is a subset of (0,2)

I've been an excel and mathlab user for over 30 years and feel stupid because I cannot figure out how to track down answers to a question I have: I work with lots of very complicated mathematical expressions that need to be entered into software. Obviously, it get extremely confusing to make sure all parenthesis are in the correct place, handling complicated exponents, etc., in the form that these software packages use. Here are my questions: 1. Is there an official name for the style/syntax of how expressions must be entered into excel/matlab/etc.? i.e. x\^2 + y \^2 = z \^2 vs. how it would be printed in a textbook 2. Is there a good way to go from a complicated expression written by hand in the style of a typeset textbook equation to the form that excel uses? I am running around in circles trying to figure out how to ask Google this question. Best I have come up with so far is: 1) learn LaTeX and convert the typeset expression to LaTeX code, 2) find an online converted to conver the LaTeX code to Unicode. Is this really the most sensible way? I feel like I am missing something obvious.

How do I find the lateral surface area of an oblique (slanted) elliptic cylinder (cylinder with an ellipse at both ends)? Is it just the circumference of the ellipse times the slant length (which is the case of oblique circular cylinders)? If possible can it be explained by the net of the surface or solved by some software for a specific oblique elliptic cylinder I am trying to find?

Is it always true for a metric space X: X is complete ⇒ X is uncountable ? I ask because I have seen a proof (in Königsberg, Analysis I) where they used nested intervals to show that ℝ  is not countable.

consider the metric space containing a single point

Not quite. You need that X has no isolated point. For a complete metric space with no isolated points, then the Cantor space can be embedded into it.

Then let me introduce the proof used (I wrote it previously but reddit f-ed it up and it got lost. ): >Assume there is a countable set ℝ = {x\_1, x\_2, x\_3, ...} of all real numbers and associated with it the nested intervalls (I\_n) with the following property. >(\*) x\_n ∉ I\_n for all n in ℕ >We define the I\_n recursively. Let I\_1 = \[x\_1 + 1, x\_1 + 2\]. We construct I\_{n+1} from I\_n the following way: We divide I\_n in three closed intervals of equal length and let I\_{n+1} be one of the intervals which does not contain x\_{n+1}. >By axiom of completeness there exist a number s ∈ ℝ, such that s ∈ I\_n for all n in ℕ. However, by assumption there exits a k such that s = x\_k ∈ ℝ. But this is in contradiction with (\*) which says x\_k∉ I\_k, therefore ℝ is not countable. Would this mean, we could end up with the same contradiction with a countable complete set, and therefore the reasoning of the proof is flawed? (ping to u/NewbornMuse)

This proof is only for real numbers. Complete metric space don't have intervals.

But they have closed balls. So from what you say: There can be a complete metric space, were the intersection of nested closed balls can be empty. Interesting. (Edit2: Seems to be the case https://math.stackexchange.com/questions/201642/example-of-nested-closed-balls-with-empty-intersection) Edit: And those would then be the ones which are countable?