are shapes finite?

[angry bob]

It is zero. Any finite number divided by infinity equals zero. There is no such thing as "infinitely small". Such a thing would be represented by 0.0 followed by an infinite number of 0s followed by a 1. But you can't put a 1 on the end of an infinite list because infinite lists don't have any end to put it on.

If two numbers aren't equal, there must be some other number in between them. But there can't be any number between zero and "infinitely small", or that number would itself be "infinitely small".

'infinitely small' is not a number. Its (I think) the limit of a number as it tends to zero.

The whole of calculus is based on this premise. i.e. anything divided by zero is infinity. But the function sin x / x as x tends to zero is one.

I actually agree that there is probably no such thing as infinitely small and hence no such thing as infinity ... but much of science is predicated on the opposite.

If space is continuous ... and you take a meter and cut it in half an infinite number of times, what you are left with is an infinitely small distance. That is not the same as no distance at all (which you would no longer be able to cut in half)

 

[angry bob]

I'm not arguing any such thing - just trying to ram the point home that an infinite set doesn't automatically imply that it contains all possible things.

hmmm ... the key here is the word 'possible' ... and what you mean by that.

 

[Kameron]

That's all very well on paper or with mathematics ... but you cant really understand it. At least, you cant visualize it. Or I cant anyway!

Visualize may be not, conceptualise, certainly. Between the moving orthogonal to a sphere, moving orthogonal to a infinite flat plain and of course the reverse of that, orthogonal to a line, I don't see it as hugely complex to imagine higher spatial and time cords. I run into serious problems with that folded dimension stuff. Ok I can deal with the 3D one sided shapes (Kline Bottle) but that is about the limit, I don't seem to be able to extrapolate from that point and lack any decent example of folded time.

Understand it, fleetingly, but I do recall magical moments of clarity though they tended to arrive in bed rather than the exam hall.

 

[Kameron]

hmmm ... the key here is the word 'possible' ... and what you mean by that.

Obviously if I create an infinite set of integers I don't expect to find a copy of the Mona Lisa if I go on counting long enough. However, if I get an infinite set of letters of the alphabet I could legitimately expect to find the works of Shakespeare reproduced in there. In fact I could also calculate how long I would, on average, have to search for.

For instance, to find kameron assuming my set contains just 26 lower case letters and I can search at a rate of 100,000 letters a second then I would expect to find those letters in that sequence on average after about 22 hours. You just need to calculate how improbable it is. In an infinite set that could possibly contain what you are looking for it will be there, you just won't have enough time to find it. Suppose that my set had upper, lower case and space and I am looking for Kameron as a word and we are suddenly up to 1046 years.

 

[laptop]

'infinitely small' is not a number. Its (I think) the limit of a number as it tends to zero.

Which is where I get to mention the surreal number system

Integers: -∞...-2, -1, 0, +1, +2...+∞

Reals: infinitely many of them between 0 and 1 and 1 and 2...

Surreals: infinitely many of them between each real - and infinitely many infinitesimals, which are actual distinct numbers. In this system.

There's a whole bunch of Fields Medals waiting out there for people who make progress toward doing calculus over the surreals...

 

[bluestreak]

i *heart* surreal numbers.

they're teh rock0r

 

[Fez909]

Do surreal numbers actually exist? Wouldn't they just be a subset of the real numbers?

Actually, they wouldn't exist. How can you have a number between an infinite set of continous numbers?

We can 'get between' the integers as they're discrete, but the real numbers are definitely not.

 

[laptop]

Do surreal numbers actually exist?

What does it mean for a number to "exist"? Do you mean in a Platonic sense?

The number system certainly exists. As does, at another extreme, the "finite field" number system that contains 0, 1, 7, 666 and no others.

Wouldn't they just be a subset of the real numbers?

They're a superset. As I said, AIUI there are an infinite number of surreals between any given pair of

 

[Fez909]

What I was getting at is that the integers and the real numbers combined form the infinite number set...so why do we need another term to describe something which already has been described?

 

[laptop]

What I was getting at is that the integers and the real numbers combined form the infinite number set...so why do we need another term to describe something which already has been described?

Nope.

The integers form a countable infinity.

The reals form an uncountable infinity.

These are very different infinities

One of the more interesting thing about the surreals is that they are a number system in which infinitesimals are actual numbers.

You can do arithmetic on them. The price you pay for this is that calculus doesn't work over the surreals. Yet, anyway. As I said, there are Fields Medals waiting for people who make it work - or rather, I suspect define something "more fundamental" than calculus that does work over the surreals.

The constructor - the bunch of logical statements that define any number - is rather elegant too, but I don't understand it well enough to describe it tonight.