Interesting Links for 17-07-2012

[andrewducker]

• How Emma Sky went from anti-war academic to governor of one of Iraq's most volatile regions
  (tags: iraq war peace )
• Screen Display Calculator - how far away should your TV be? And what size?
  (tags: tv monitor size calculator )
• Eight radical solutions to the childcare issue
  (tags: childcare children government )
• What did the Persians think of Alexander the Great?
  (tags: history persia middle_east greece )
• Free access to British scientific research within two years
  (tags: science research uk epicwin # )
• 7 reasons why I’m against DevoPlus / DevoMax
  (tags: independence scotland uk )
• Merely visiting a newspaper website can be a breach of copyright.
  (tags: copyright wtf uk law )
• How many infinities are there?
  (tags: infinity )
• The three options the Church Of England faces over same-sex marriage (well worth reading)
  (tags: ChurchOfEngland lgbt marriage uk religion )
• Fifty Shades of Babe: Duke Nukem Reads E. L. James
  (tags: video funny porn dukenukem )
• Facebook Engineer Responds to Imgur Block with Epic Reddit Apology
  (tags: facebook images reddit funny )
• US Security Agents At Heathrow For Olympics
  (tags: security usa uk )

Comments

How many infinities are there?

[cartesiandaemon.livejournal.com]

ROFL. That's really well written. I guess one might guess the answer is "infinitely many!" but it's good to see it explained.

(And as far as I can tell accurate: it's scary to think that now I know fewer practicing mathematicians, I probably know more about infinities than almost all of my friends :))

For the record, "infinity" is more of a label mathematicians and non-mathematicians slap on stuff which is "too big to count", so it usually means ordinals and cardinals, but there are some other uses.

For instance a number is often adjoined with a +INF and -INF which work roughly the way you'd expect (INF>x for any finite x; INF+x is INF for any finite X; INF-INF not defined). There are two of those. Eg. floating point representations on a computer often have somehting like this with a special bit pattern for results which are +INF, NAN, etc.

For complex numbers, there's only one INF, which is sort of "round the edge" of the whole plain.

Although obviously, if you lumped those in with the ordinals and cardinals, the answer would still be "too many to represent" :)

Re: How many infinities are there?

[cartesiandaemon.livejournal.com]

And also for the record, there are alternative formulations of set theory where you don't have to put up with the "proper classes are things which are exactly like sets, but are not sets because that would cause paradoxes" stuff, but you have to give up something else you would expect (eg. the ability to say "the set of all elements of set x which are 'some property'")

But everyone uses the normal one, and I don't think the alternatives have proposed anything significantly better.

[spacelem.livejournal.com]

If I were writing an essay about infinity, I'd say that while there are a lot of infinities, there are really only two kinds of infinity that you need to worry about: countable and uncountable.

Countable infinity is when you can order all the items in a set and label them 1,2,3,.... . You'll never end (obviously, because there are infinitely many of them), but every item in that set has a label. The set of integers is of this sort, because every integer has a name. We call this infinity aleph-null (א₀).

Uncountable infinity is when you can't do that, because there are too many elements -- it's not clear what label something should have. For example, 0.999... with countably many 9s is exactly equal to 1, so two numbers with different labels are the same. What's more, is that with uncountable infinity, no matter how close two numbers, there are an uncountably infinite number of numbers between them. This is true of real numbers.

I think that's a rather more useful explanation of infinity, even if incomplete.

[khoth.livejournal.com]

Your examples there aren't great - the rational numbers have both the things you cite for uncountable infinity: Labels aren't unique, eg 1/2 = 2/4, and between any two rational numbers there are an infinite number of other rational numbers. But the rational numbers are countable.

The thing with labels isn't to do with something having more than one label, it's about there not being enough *finite* labels to include everything. So 0.999.... isn't even acceptable as a label. The problem with real numbers is that there are uncountably many of them that just go on and on with no actual pattern, so you can't describe precisely what they are.

[spacelem.livejournal.com]

The examples were correct -- integers are countable, reals are uncountable. The problem was in the explanation of the labelling part for the uncountable numbers, but I was in a bit of a rush and didn't think too carefully about it (this is me time wasting while I'm thinking about cover letters for a paper and a job interview).

The rationals are easy to label: have a table with the integers on top and side, generating rationals by top/side (and being sensible about 0), then move diagonally back and forth labelling all the numbers. There will be some overlap, but it's fine to give the same number two labels when it can be presented in two different ways. There are, however, only countably many rationals between any two numbers, not uncountably many.

The take home message remains that there are two types of infinity that should suit most people.

[khoth.livejournal.com]

Sure. But your "uncountably dense" condition is a bit of an odd thing to stress, since it's somewhat circular, and isn't a necessary condition for a set to be uncountable (eg the cantor set and ω₁ aren't dense everywhere), and normal desnseness isn't sufficient for uncountability.

[spacelem.livejournal.com]

I'm an applied mathematician, not a pure one (not since undergraduate at any rate), and preoccupied with other things. It wasn't particularly precise, or all-encompassing, but it got across the two important infinities that most people are likely to come across.

[andrewducker]

I'm not sure I'm likely to come across any infinities, most of the time (short of, well, looking up. I hear space is big.)

[danieldwilliam.livejournal.com]

Hey, that's why they call it Space.

:-)

[andrewducker]

Was that an Under Siege 2 reference?

[danieldwilliam.livejournal.com]

More Hitchhikers Guide to the Galaxy than Under Siege 2.

[andrewducker]

I originally thought it might be - but I can't find a reference to that line being in the books!

[danieldwilliam.livejournal.com]

It's not a direct quote. More in the spirit of HHGTTG.

[andrewducker]

That'll be my awful geek upbringing, expecting people to have memorised entire books/TV series/movies so they can quote them at just the right moment!

[danieldwilliam.livejournal.com]

It is to my shame (oh, the shame, mother, the SHAME!) that I couldn't remember the exact quote.

Geeks FTW.

[andrewducker]

Some day the pain will go away. When you have the internet in your brain, and can thus remember the perfect quote instantly.

[cartesiandaemon.livejournal.com]

Or (arguably) dividing by zero :) Or (even more arguably) fielding questions from teenagers about "Are you SURE 0.99999999999 is not less than 1"? :)

[andrewducker]

Pffft, division by zero is undefined, everyone knows that :->

[cartesiandaemon.livejournal.com]

Oh yes, but sometimes people find it's more useful to distinguish between infinity and other sorts of undefined :)

[steer.livejournal.com]

The cantor set is countable surely?

[khoth.livejournal.com]

No, it's uncountable. For any real number between 0 and 1 (of which there are uncountably many), you can express it in binary eg 0.0010101101..., then replace each 1 with a 2 to get a ternary number eg 0.0020202202... which is in the Cantor set.

(This glosses over some technical details about numbers with two binary representations but that doesn't make much difference)

Incidentally, Greg Egan once wrote a short story where it was critical to the plot that the Cantor set was uncountable. I'm pretty sure he did it to prove he could.

[steer.livejournal.com]

You are absolutely correct. Indeed, now you mention it, I can recall the very same discussion the first time I heard it about 15 years ago with two number theorists in a ground floor office in York University while they minded the dog belonging to a retired professor of pure mathematics. I think it's the only impressive proof I remember (or I remember now you remind me) involving ternary.

When it comes down to it the properties of the Cantor set are quite remarkable... measure zero, nowhere dense, a complete metric space yet uncountable. No wonder it gave mathematicians of the age fits!

I mentioned elsewhere (not sure if in reply to you) Rudy Rucker's book "White Light" which also hinges on the countable, uncountable and other possible forms of infinity.

[steer.livejournal.com]

Surely Mr Spacelem you must at least address the continuum hypothesis if you're counting the infinities.

I would answer "there are *at least* two infinities" -- the ones you state (aleph nought and c). We do not know if aleph one is the same as c -- and it may require extra axioms in our chosen set theory.

http://en.wikipedia.org/wiki/Continuum_hypothesis

Incidentally, Rudy Rucker's terrific but weird "White Light or What Is Cantor's Continuum Problem?" addresses the issue in a scifi way with lots of drugs and giant talking cockroaches.