@DavidWheeler Gluing seems less coarse.

Sewing hurts.

@robjohn Is that the guy who wrote the "lifting exponents" exposition on AoPS?

@PeterTamaroff Depends. If the glue is hot, for instance...

@PeterTamaroff an asutute observation...from what you know, can you show that this means gluing is more likely to be continuous?

@BillDubuque I don't know who it was for sure. The points lost by downvoting are smartly hidden from view :-)

@robjohn The thing I like about the telescopic view is that it allows you to view it as a product of n terms all divisible by 3, so the product is divisible by 3^n. That's about as obvious as it gets.

even a dweeb like me can understand that.

@DavidWheeler Well, glue is well behaved.... unlike thread.

facedesk

@BillDubuque That was the idea behind $(1)$ and $(3)$ in my answer.

@DavidWheeler Whaaaaaaaaaaat?

@BillDubuque I thought I tried that on myself to adjust my score, but nothing changed. Perhaps I didn't wait long enough.

ok, when i explain it, it ruins the joke, tho

@DavidWheeler Well, explain it.

topologies can be coarser or finer

@robjohn But did you really view 10^3^n - 1 as a product? I don't see that explicitly mentioned in your answer.

the finer the topology of the domain, the more like it is that any set (including a pre-image set) will be open

@DavidWheeler I'm asking about the facepalm/desk thing.

@BillDubuque $10^{3^n}-1=\left(10^{3^{n-1}}-1\right)\left(10^{2\cdot3^{n-1}}+10^{3^{n-1}}+1\right)$

facepalm u know....facedesk just hurts more

@DavidWheeler What motivated the facepalm¿

your response that glue is "well-behaved"

i thought we had something special...but...at least we have paris

@DavidWheeler I meant glue IRL

@PeterTamaroff ...but it's easier to make a mess with glue than with thread.

are you saying glue isn't order-preserving?

@J.M. But that is your fault, not the glue's! And thread knots!

@robjohn Yeah, but one hopes the reader can make the conceptual leap from that term ratio to the entire product, which is why there is a name for (multiplicative) telescopy. But I agree, the essence is there, even if the explicit structure is not.

knot theory is a very tangled subject

@DavidWheeler bad-dum tss

my gf asked me the other day: "what are you reading?" i said: "the mathematics of hair arrangement" (as i was reading about braid groups). i think she thought i was kidding.

@DavidWheeler It certainly is a subject that's easy to get tied to...

kind of kinky, in fact

@DavidWheeler Loopy, also.

@DavidWheeler I had a lot of fun with that and old gf when I was studying braid groups.

it amazes me that even the simplest things, can produce questions of great complexity

@DavidWheeler You can't get much simpler than the intractable hailstone 3n+1 problem.

Goldbach's conjecture is pretty simple, too...

So is it a failure on our part to recognize some aspect of complexity?

replace "aspect of complexity" with "complex aspect" and I'd say that's a good way to put it

More of a failure of layperson logic - that simply stated problems should have simple solutions.

oh...that has a different name....

collatz conjecture, that's it

@BillDubuque Put that way, then it's false for a lot of things. Not just math.

@DavidWheeler yes, but "hailstone" is quite picturesque...

simple systems can produce complex behavior

this give me hope that the world we live in may in fact have simple rules

I don't see why we can't conceive of a system based on its behavior in the first place

Or rather, why we don't ...

as laypeople

@CLarue i'm not sure what you mean by that

If we think of a some proof of a theorem upon reference to its name, then we don't have a false sense of simplicity.

Of course, I see why that is intractable, so my earlier statement is still hogwash.

@DavidWheeler But the point of 3n+1 is that even though the rules are simple the behavior can be very complex. Congruential iterations like 3n+1 can be programmed in very small Turing machines. Generally they are undecidable (Conway). Being so simple, these could be coded in nature somewhere by evolution.

well a great deal of work has been to done to recover substantial parts of mathematics based on the sheffer stroke...which is to say, we can build powerful computers out of nothing but NAND gates

@BillDubuque yes, and the rules for Conway's "Life" are also simple, as is Rule 110

or: i understand the rules of go completely, but i barely understand the game even a little

@DavidWheeler Ah, automata. That's what Stephen Wolfram's been going crazy about in recent years...

in fact, it is my understanding that the mandelbrot set has a "picture" of the rationals, scaled by the size of the denominators...in the "bulbs"

just the recent ones?

Stephen Wolfram was quite sane about automata in the 19th century

@anon Well, I count the latter part of the '80s as "recent"... :D

i feel it safe to conjecture the same was true in all previous centuries, but the margin is too small to prove it

ah, the '80s.....the last time i actually owned a radio.....

do those still exist?

@DavidWheeler You don't listen to the radio anymore?

when was the last time you owned a ... car?

2007, i think

I was being rhetorical. (cars have radios)

mine did not...only a CD player

Hell, I don't have a car anymore, but I still listen to the radio...

last year, i got rid of my televison, too...heck my monitor is bigger

Not too long till one can say the same for physical (paper) books.

10 years ago, i had an idea which i should have tried to patent

a flexible lcd screen, the size of a newpaper page, with a data port

@DavidWheeler Hopefully e-paper will be mainstream soon ... lots of interesting prototypes recently.

cpus and data storage are smaller than fingers, now

@BillDubuque That word "soon"... it takes too long sometimes.

@J.M. But soon is sooner when one is older... times passes faster.

for example: nearly a decade has passed since we began this conversation.

@DavidWheeler Are we all that old?!

i don't know about you, Bill, i am 51.

the time when i was studdying math in earnest is long past...now, i just do it to keep my mind from decaying...as fast.

@DavidWheeler We're fairly close then. I vaguely recall your name from the past, perhaps sci.math or another forum? Or MIT?

age-wise, perhaps. :P

perhaps another forum...

but..i wouldn't put too much stock in that...Wheeler is a common name, as is David

but...I am in neither movie....although, I have struggled with mental illness in my life

wheeler reminds me of joey wheeler from yu-gi-oh. im so ashamed.

@J.M. Ooh, that's bad. Ok. Thank you!

One more vote needed.

Reminds me, I'm due for a molt...

@J.M. Nooo...!

@J.M. Someone tells me there are newer ones that have less negative effects. I'll go for those I guess.

@MattN. More expensive, though.

@MattN. Oh, you prefer the fractal?

Good morning

@J.M. Well. I cannot answer this question, can I, since I don't know what you were going to swap it for. But what you could so is change its Salmon colour to something like I nice green or blue. : )

@OldJohn Good morning.

the contrast value of its color is not so high

@MattN. I'll think about it... :)

perhaps a beltrami pseudosphere....

@DavidWheeler The base has a bit more contrast, yes.

@DavidWheeler I did a more elaborate version of it a week ago... let me get the picture.

that is sooo pretty

It's called the "breather" pseudosphere. Slightly more elaborate, but still has constant negative Gaussian curvature.

i used to have as my desktop pictures of surfaces of constant negative curvature that had been photoshopped to look as if they were made from blown glass

Ah, I can imagine how pretty they look.

i forget where i found the image...but they were part of a competition for mathematically based art

@Jas

Why do we close questions that are exact duplicates?

@JasperLoy Thanks - felt it was time for a change - and that fractal fascinated me years ago

@BenjaLim I assume it makes more sense to merge them in some cases, and makes the site more sensible as a repository of answers

@JasperLoy It can be a bit annoying to write an answer to a question that then gets changed, and then having an answer downvoted (only happened to me once, I think)

@JasperLoy I thought deleting it would just be better

@JasperLoy my algebra course is based on brian hall's book

yes

@JasperLoy know much about the book?

I just ordered it from amazon

@JasperLoy what do you think?

Our lecturer is jarod alper

@JasperLoy No i meant about rep theory

today

yes

@JasperLoy what kinda music you like?

that is the course text, brian's book

and jarod's an AG guy, seems pretty pro

@JasperLoy what's with all this going on with JB?

@JasperLoy youtube.com/watch?v=KowDzt31jUM

@JasperLoy but that is of finite groups yes?

@MattN. hey yoyoyoyoyoy

@J.M. Oops, sorry for all the typos. Not sure how I managed to make so many in such a short message. : )

Bbl

I have studied some on finite groups before :D

I did a thing to classify all irreducible reps of $S_4$

tes

Meow.

We need more crazy cat ladies.

@JonasTeuwen Good morning

Good morning!

At work :-). Had nice breakfast in the center. Nobody here... 8-). PERFECT.

@JonasTeuwen It is also pretty quiet here (I am alone in the house) - and also here in chat it is quiet :)

@OldJohn Hmm :-). Oh, I just saw a graduate student I think...

They are nicely tucked away in a small office.

I have a much bigger one 8-) (while I should be in the small one).

@JonasTeuwen Best keep them out of sight :)

Yep.

hi

about math.stackexchange.com/questions/174110/is-every-set-a-subset

I can't add comment to the answer

why?

You don't have enough rep.

@Ankush You need at least 50 reputation to add comments everywhere

can you add following comment to the answer on my behalf?

Answer says: Yes, one can: S∪{S} is a proper superset of S, since S∈S∪{S}, but S∉S. Thus, S⊆S∪{S}, but S≠S∪{S}.

comment: Why is this S≠S∪{S} true ?

What comment did you want to add to that?

mentioned above

thanks :)

The author already explained why $S\ne S\cup\{S\}$; the latter contains $S$ but the former does not, and since two sets are equal if and only if they share the same contents, the two sets must be nonequal.

@Ankush $S\notin S$ follows from Axiom of regularity

(Which I mentioned in a comment to Brian's answer.)

yes... just read about it on wikipedia

I don't get this axiom

consider a finite set {a, b, c }

how can you apply this axiom here ?

However, if someone learns naive set theory and not axiomatic theory, then Asaf's answer is more accessible.

here S≠S∪{S} is false. right ?

where S = {a,b,c}

It depends on what is a,b,c.

S is a set of english alphabets

26 count

now?

or S is a binary set

0, 1

This question is, to some extent, related: When is $x=\{x\}$?.

If S={a,b,c}.

Then S∪{S}={a,b,c,{a,b,c}}.

So it is a different set.

yes

got it

thanks :)

@PeterTamaroff That is like telling me to run my petrol car on diesel to save the environment (for the record: I do not own a car, we do car sharing)

Without coffee I would cease to be.

Mm... you tell that to a (semi-)professional Barista? 8-).

(I have plenty of tea and many methods of making it).

Moreover it comes from an obscure Chinese dialect most English speakers have never heard of. (I wonder how that happened?)

hey

@BenjaLim Hi

@ZhenLin apparently you can blame the Dutch, or more specifically the Dutch East India Trading company.

and i've heard of Hokkien, but only because someone i knew from singapore said it is often used for swearing as it is considered more "rough" than Mandarin (or so i was told)

@J.M. I received Don Knuth's letter.