It just occurred to me what was the meaning of your challenge to "find" the powerset. I think you are making the point (no pun intended) that if we have six points that form a set (as the nodes of a graph do, per the Wiki article on graph theory), you are saying that without more information, I could not find the powerset. You are actually correct on this subtle philosophical point. (Not in the case of a finite set, but in the general case). The thing is, I don't have to find them. They exist by virtue of the powerset axiom. https://en.wikipedia.org/wiki/Axiom_of_power_set That is an axiom of set theory that says that if you have a set, its powerset exists. But note that is is a very powerful axiom that posits the existence of things we could never find: namely, ALL the subsets of a given set. For example take the set \(\mathbb N = \{0, 1, 2, 3, \dots\}\), the familiar set of natural numbers. By the powerset axiom, the set \(\mathscr P(\mathbb N)\) exists. Its elements are ALL the subsets of the natural numbers. We know from the time of Cantor that the powerset of the natural numbers is uncountable. What does this mean for our ability to "find" each of the subsets? Well, if we define "finding" a subset as writing down its elements, or writing down a formula or procedure or computer program for generating all its elements, then most subsets of the natural numbers are not findable! That's because there are only countably many Turing machines, computer programs, or finite-length descriptions of sets. So all but countably many subsets of the natural numbers are unfindable in this sense. Such subsets are officially called noncomputable. But we know they exist. Why? Because the axiom of powersets says so. And this is a point of departure for mathematicians who believe in constructive math. They would say that ONLY those sets exist whose elements can be cranked out by a Turing machine. So there's a lot of serious philosophy lurking here. https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) Perhaps this is the subtle issue that you were getting at when you asked me to find the powerset of a six element set. For finite sets that's no problem. But in the general case, it's impossible. Nevertheless, the powerset exists. Where does it exist? In Platonic heaven with all the other mathematical objects.
Yes, but what does that have to do with anything we are talking about? That's not an argument, it's not a conversation. It's another change of subject.
I'll try to go over what, at least what I think, I've been talking about, whatever it is that you think you've been talking about.
If you keep it simple and stick to one little subtopic at a time, progress might be made. You did convince me about the triangulated sphere. I don't doubt you have interesting things in mind. But since graph theorists are using tensor products in a different way than algebraists do (as far as I can tell) I no longer have confidence in that calculation of the iterated tensor products. It's correct algebraically but I don't know how graph theorists regard it.
No, the subject initially was the Riemann sphere, the projective plane, Platonic tilings, reflection groups, and a lot of stuff about a graph that I have. In fact you've been changing the subject, but I'm not that concerned. What does concern me is your apparently poor grasp of graph theory, and graphs are pretty basic things, albeit there's a lot of theory. You "correct" me repeatedly about the fact that all the groups I'm talking about, the group product I've notated with perfectly reasonable notation as <2,3>, is actually \( Z_6 \). Therefore you seem to also be saying the graph can't exist, there is no copy of this <2,3> group product, because it's bloody well is actually \( Z_6 \). That's what I have a graph of . . .?
The bloody train is bloody late You bloody wait, you bloody wait You're bloody lost and bloody found Stuck in fucking Chickentown The bloody pies are bloody old The bloody chips are bloody cold The bloody beer is bloody flat The bloody flats have bloody rats The bloody clocks are bloody wrong The bloody days are bloody long It bloody gets you bloody down Evidently Chickentown
That's the first thing you've said that makes a lick of sense to me. I had to look it up and it's a very cool song.
Yes, it's really what you could also dub a topological invariant--the powerset is there without needing to characterise it.
Someguy1 has obviously been studiously correcting me about the sameness (isomorphisms) of certain groups. I've been trying, just as studiously to correct him about his grasp of drawing graphs that then can be given a group structure. Maybe someone will submit a paper proving that a graph of six decorated points on the same line is isomorphic to a graph of two lines with three decorated points on each line. If someguy1 manages it, graph theorists around the world might well marvel at it, after they all stop laughing. And that particular subject, a proof that two graphs are the same graph because each can be given the same group structure, is the train we're waiting for. It isn't coming though. The bloody train is bloody late You bloody wait, you bloody wait . . .
Here for example, an objection to pretty much everything I've posted up to the time this is posted: Someguy1 seems to be trying to "correct" me about graph theory, which says, yeah sure, there is only one group of order 6. Big deal, there are many different (non-isomorphic) graphs that embed that group. How many? As many as you need. So it seems someguy is projecting his own confusion, misunderstanding of graphs, and confidence with his own knowledge of groups, onto me. Me!? What the hell did I say? I would seriously like to see him tell a graph theorist they're using the wrong notation.
Cars collide, colours clash Disaster movie stuff For a man with a Fu-Manchu moustache Revenge is not enough There's a dead canary on a swivel seat There's a rainbow in the road Meanwhile on Beasley Street Silence is the code . . .
Oh well, here's the rest of Beasley Street: take it away Jonno. (sorry about the relentlessly dark tone, the cynicism, etc, but I didn't write it) Far from crazy pavements, The taste of silver spoons A clinical arrangement, On a dirty afternoon Where the faecal germs of Mr Freud Are rendered obsolete The legal term is "null and void", in the case of Beasley Street In the cheap seats where murder breeds, Somebody is out of breath Sleep is a luxury they don't need, A sneak preview of death Belladonna is your flower, Manslaughter your meat Spend a year in a couple of hours, On the edge of Beasley Street Where the action isn't, That's where it is State your position, Vacancies exist In an X-certificate exercise Ex-servicemen excrete Keith Joseph smiles, And a baby dies in a box on Beasley Street From the boarding-houses and the bedsits, Full of accidents and fleas Somebody gets it, Where missing persons freeze Wearing dead men's overcoats, You can't see their feet A riff joint shuts, opens up, Right down on Beasley Street Cars collide, colours clash, Disaster-movie stuff For a man with a Fu Manchu moustache, Revenge is not enough There's a dead canary on a swivel seat, There's a rainbow in the road Meanwhile on Beasley Street, Silence is the code Hot beneath the collar, An inspector calls Where the perishing stink of squalor Impregnates the walls The rats have all got rickets, They spit through broken teeth The name of the game is not cricket, caught out on Beasley Street The hipster and his hired hat Drive a borrowed car Yellow socks and a pink cravat, Nothing la-dee-dah OAP, mother-to-be, Watch the three-piece suite When shit-stoppered drains And crocodile skis, are seen on Beasley Street The kingdom of the blind, A one-eyed man is king Beauty problems are redefined, The doorbells do not ring A lightbulb bursts like a blister, The only form of heat Here a fellow sells his sister down the river on Beasley Street The boys are on the wagon, The girls are on the shelf Their common problem is That they're not someone else The dirt blows out, The dust blows in, You can't keep it neat It's a fully furnished dustbin, 16 Beasley Street Vince the ageing savage Betrays no kind of life But the smell of yesterday's cabbage, And the ghost of last year's wife Through a constant haze Of deodorant sprays, He says retreat Alsatians dog the dirty days, down the middle of Beasley Street People turn to poison Quick as lager turns to piss Sweethearts are physically sick, Every time they kiss It's a sociologist's paradise, Each day repeats On easy, cheesy, greasy, queasy, Beastly Beasley Street Eyes dead as vicious fish Look around for laughs If I could have just one wish, I would be a photograph Of a permanent Monday morning, Get lost or fall asleep Where the yellow cats are yawning Around the back of Beasley Street
Where have you been? I first heard this in the 70's at high school, we had an English teacher who was into poetry, lyrics, etc. You know, James K Baxter, Leonard Cohen, Bob Dylan. Would it surprise you to hear I've never heard of The Frogger?
Who's that? You've never heard of him?? Are you telling lies again??? Who are you talking about??? Why is it such a great offence to you that I've never heard of something that you HAVE heard of?? I imagine there are things I have heard of that you haven't!!!
I'm trying to assume here, that you aren't being serious? Otherwise why imply I'm offended? Offended by what? By life?
You can all chill folks, I've decided to voluntarily admit myself to a math rehab clinic. (Just kidding). I apologise in advance for the fact I have two degrees, one in IT, and for knowing a bit more than the average Joe Lunchbox about the problem I'm trying to solve. I could try with a little help, but I want it from my friends. So I'm probably going to just bang on like I usually do, if you get lost because I seem to be inventing a new language, that's not my problem. I may respond to one or two objections, unless I see how pointless and off-topic they are. So, let's talk about graphs. Forget about groups, forget about fields, forget about algebra. A graph is a topological space, so there is no geometry (there is maybe, a pre-geometry). There is no notion of distance or direction, but there is "separation", and there is "adjacency". Adjacency isn't a big problem (to define): any two points connected by an edge are adjacent. The fact that an edge "does this" to two points means the edge is the separation. There is still no notion of a length for edges, and no direction defined. It therefore does not qualify as a metric space, but that does not mean it can't be given a metric; if it can, the metric does not have to correspond to a Euclidean distance. Moreover, graphs can be added to graphs, and there are several ways to define a product of graphs (you might call one of them a tensor product, but then find out there's a clash of notation). Similarly, there are various products of vectors and matrices. The inner product, cross product, outer product, and yeah, tensor product. K?