Mathematics

So suppose $\kappa$ has cofinality $\lambda>\omega$, the set of cofinality $\omega$-th ordinals is reflected with the set of cardinals with countable cofinality below $\kappa$.

So if SCH holds for all these limit cardinals, it has to hold for $\kappa$.

@robjohn Everyone's a critic nowadays. It's a long list.

robjohn

@AsafKaragila I found myself down the list, but I just got the badge. And I still don't think I've downvoted anything.

Asaf Karagila

@robjohn Indeed your profile agrees that you have never downvoted.

robjohn

I may have mistakenly pressed the down arrow before undoing it and pressing the uparrow.

Paul Slevin

@AsafKaragila "@PaulSlevin Well, you know that the behavior of singular cardinals of uncountable cofinality is determined by a stationary behavior below it." Do I know this

I have Silver's theorem from Jech's book and a lemma by Prikry that I used to solve it

Asaf Karagila

Well, this is Silver's theorem.

robjohn

12:08 AM

I hope they don't count a clicko

Paul Slevin

I have this "Let $\kappa$ be a singular cardinal such that $\operatorname{cf}\kappa > \omega$. If $2^{\alpha} = \alpha^+$ for all cardinals $\alpha < \kappa$, then $2^{\kappa} = \kappa^+$.

Asaf Karagila

@PaulSlevin Yes, but it is actually enough to require a stationary set.

Paul Slevin

so its enough to have $2^{\alpha} = \alpha^+$ on a stationary set, yes?

but that is the GCH not the SCH

or does GCH $\implies$ SCH

Asaf Karagila

Of course that GCH implies SCH.

It is a theorem of ZF(C) that $\kappa<\kappa^{\operatorname{cf}(\kappa)}\le\kappa^\kappa=2^\kappa$. Under GCH $\kappa^+=2^\kappa$.

You also have the Galvin-Hajnal theory which tells you that it is not only $2^\alpha=\alpha^+$ but rather any behavior of the continuum function on a stationary set will be reflected to the limit.

Paul Slevin

yeah but if SCH holds for the cardinals with $\operatorname{cf}\alpha = \omega$, does that mean that $2^{\alpha} = \alpha^+$ ?

Asaf Karagila

12:15 AM

SCH says that $\kappa^{\operatorname{cf}(\kappa)}=\kappa^+$.

Paul Slevin

But how can I use that given what I have - I would need $2^{\alpha} = \alpha^+$

oh man. I think I am missing a lot of information here

Asaf Karagila

@robjohn: Seems that I have endowed you with a silvery badger as well.

robjohn

@AsafKaragila It would seem that I am so enlightened.

@Asaf: you still need more badges ;-)

Asaf Karagila

12:30 AM

:-P

robjohn

OMG... Brian Scott is ahead of Arturo for the week !

Asaf Karagila

Give it a few minutes :-D

robjohn

True, it's only 30 points

@AsafKaragila I've only been enlightened 3 times. I appears that you have received enlightenment 22 times.

Asaf Karagila

23 now.

robjohn

@AsafKaragila Ah the one I voted for hasn't shown up then.

just did

Asaf Karagila

12:39 AM

Now it does.

Heh, I voted 1,999 answers.

robjohn

I'm a measly 384

@AsafKaragila that includes up and down votes, right?

Asaf Karagila

Yes.

Well, now you have one more badge; your first answers page is all Nice Answers; and I have 2,000 answer votes.

Paul Slevin

soreeee me again

is $\kappa^+ \ge \kappa^{\operatorname{cf}\kappa}$ always?

Asaf Karagila

No.

If anything $\le$.

robjohn

@AsafKaragila Ah, if I sort by votes :-)

I get an error on the reputation page

Asaf Karagila

12:46 AM

I'm good.

robjohn

@AsafKaragila why do you say that?

Asaf Karagila

I don't get an error on my reputation page. :-)

robjohn

same link?

Paul Slevin

Thanks

Asaf Karagila

@robjohn Yes, it's an individual page, you know.

robjohn

12:49 AM

I know, but I wonder why I get an error.

robjohn

1:00 AM

Hmm. Somehow I was logged out on that page.

user19161

1:24 AM

No need to do rep recalc anymore!

Asaf Karagila

Well, screw that. I am going to sleep.

Paul Slevin

goood night

user19161

Please refresh to see my new avatar!

user19161

1:40 AM

@robjohn You know nobody can see this except you right?

user19161

If we click there we will just see our own when logged in or nothing if not logged in.

user19161

Over and out!

Rajesh D

2:44 AM

Hey !...the room seems dead and asleep !

user19161

@RajeshD Boo!

user19161

Of course the room is dead. But we are alive.

user19161

The room is not a living thing.

user19161

And it's Fri night, so the cool kids are out partying.

robjohn

2 hours ago, by robjohn

I know, but I wonder why I get an error.

2 hours ago, by robjohn

Hmm. Somehow I was logged out on that page.

@Jasper: I thought that if I was getting an error going to that URL, perhaps something was broken. When Asaf said that he did not see an error, I figured it had something to do with my account.

Rajesh D

2:55 AM

just a few ghosts here and there may be !

robjohn

@Jasper: However, it was simply a matter of bad cookies, I guess. I noticed that on the error page there was a link to log in. I logged in and everything was fine. What is odd, is that Firefox had my cookies right in the other windows.

user19161

@robjohn There have actually been a number of questions on various metas on what info is visible to the public. The simple answer to all of them is just log out and whatever you still see is the publicly available info, but a lot of people still don't know this I think!

robjohn

@RajeshD ghosts?

@JasperLoy I figure that if I can see it on someone else, they can see it on me :-)

Rajesh D

How can i be sure i am talking to real people, may be there are some ghosts sitting in my laptop and doing the talking

user19161

@RajeshD You can never be sure because real people and ghosts are isomorphic.

user19161

2:58 AM

You just need faith. I think one of the Conways once said "Truth is more important than proof".

user19161

I like to say "Truth is based on proof; faith is based on evidence".

user19161

Note that proof and evidence are distinct; so are truth and faith.

Rajesh D

okay

user19161

@RajeshD And I like to type OK instead of okay, because OK is the original form of the expression.

user19161

It is also shorter so I wonder why people like okay.

user19161

3:02 AM

Just like DJ came before deejay.

user19161

So I see skullpatrol is back!

Rajesh D

@rob : A while ago I asked @tb about some good books for Differential Geometry and he suggested a must read, good down to earth book before going onto abstraction, but i do not remember its name (although i downloaded a copy it frim internet, i don't know where it is now)....is there any good way to know it by searching the past ?

user19161

@RajeshD Was he talking about Do Carmo?

Rajesh D

yeah perhaps....yes

must be i guess

Dylan Moreland

Oct 21 '11 at 8:57, by Rajesh D

@t.b. : is it this ? http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

When everything is stored and searchable, you don't have to guess.

Rajesh D

3:04 AM

wow

user19161

Do Carmo has "Differential Geometry of Curves and Surfaces" and also "Riemannian Geometry".

Rajesh D

how to do it ? please let me know

user19161

But I suggest Kuhnel's Differential Geometry which combines the two books above into one short concise text with less topics of course.

Dylan Moreland

There's a search box in the top right.

Rajesh D

ok

Dylan Moreland

3:06 AM

If you use it, you get to a page which allows you to search by "who said it".

Which is the key.

Rajesh D

didn't know the search was so good

ok

user19161

Spivak's 5 volumes is too verbose. You may also wanna look at Lee's 3 volumes.

Rajesh D

ok @jasper

Dylan Moreland

I like Lee's books a lot but he takes along time to get where he's going.

Rajesh D

to begin with i just want to have a good grasp on the fundementals

@Dylan : what is the title of the book by Lee ?

user19161

3:09 AM

@RajeshD Introduction to topological manifolds, Introduction to smooth manifolds, Riemannian manifolds

Dylan Moreland

What I like about do Carmo is that he spends time on pretty concrete situations at the beginning.

Rajesh D

looks like a bit advanced to me

user19161

@RajeshD Take a look at Kuhnel. It's cheap too.

user19161

Translated from the German text.

Rajesh D

then i guess Carmo would be good to me and later move to others...i'll make a note of them this time though

user19161

3:12 AM

Why are math textbooks so terribly expensive?

Dylan Moreland

I think the danger with books on manifolds is that they usually become long strings of definitions.

user19161

Why are great texts out of print?

Dylan Moreland

Money.

user19161

They should make all math textbooks like open source software.

user19161

I think the above deserves to be starred.

user19161

3:17 AM

Supposedly the author doesn't earn much and the publisher earns a lot.

user19161

And then one day they decide not to publish it even though they can offer print on demand.

anon

3:46 AM

"which I assume is a quadratic" cringe

robjohn

4:11 AM

Oct 21 '11 at 8:57, by Rajesh D

@t.b. : is it this ? http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

Rajesh D

@thanks @rob I got it from @Dylan too

robjohn

@RajeshD Ah, I see it a bit below your question. I should scroll further :-)

Commutative Algebra

Transcript for

$Mathematics$ Commutative Algebra

$Mathematics$ Mathematics