# The Naked Scientists Forum

#### Judge

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« on: 03/02/2008 22:35:37 »
hey i was not to sure where to post this so sorry if its in the wrong place.

Well one time I heard from my old algebra teacher that there were two kinds of infinite. She said that a mathmaticion, whom I forget the name of, proved it. One type was that there are an infinite number of numbers and the other was that the number os grains of sand are infinte. I have been wondering who it was and what the two kinds of infinite were called, as I have forgetn what she said because it has been a number of years since. Also does anyone know why this is?

#### DoctorBeaver

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« Reply #1 on: 03/02/2008 22:43:30 »
Certainly there is more than 1 definition of infinity. I'm not sure about the grains of sane, though. That must surely be a finite number albeit extremely large.

Have a look at this site for some infinite goodies...

http://diveintomark.org/archives/2003/12/04/infinite-hotel
« Last Edit: 03/02/2008 22:48:02 by DoctorBeaver »

#### another_someone

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« Reply #2 on: 04/02/2008 03:01:49 »
The closest I can get to finding an answer to your question is:

http://en.wikipedia.org/wiki/Aleph_number
Quote
In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph.

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line. While some alephs are larger than others, ∞ is just ∞.

#### DoctorBeaver

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« Reply #3 on: 04/02/2008 07:51:42 »
The closest I can get to finding an answer to your question is:

http://en.wikipedia.org/wiki/Aleph_number
Quote
In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph.

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line. While some alephs are larger than others, ∞ is just ∞.

Yeah... that too. Erm... can anyone translate it into something this dumb ol' beaver might understand?

#### opus

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« Reply #4 on: 15/02/2008 22:42:00 »
Do people who believe in infinity ever say never......?

#### another_someone

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« Reply #5 on: 16/02/2008 01:06:03 »
The closest I can get to finding an answer to your question is:

http://en.wikipedia.org/wiki/Aleph_number
Quote
In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph.

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line. While some alephs are larger than others, ∞ is just ∞.

Yeah... that too. Erm... can anyone translate it into something this dumb ol' beaver might understand?

OK, I will try and give it a go, and hope I don't get it ghastly wrong.

The issue is primarily between cardinality and ordinality.

Cardinality is a number used to represent size - e.g. that there are 10 of something, or that there are an infinite number of things.

Ordinality is a representation of the order of things, so one refers to the 10th in order.

Aleph numbers represent infinite sized sets (e.g. if one looks at the number of even numbers, one can say there are an infinite number of even numbers, but one is not concerned about the actual order in which the numbers exist, or even what the names of the numbers are).

Ordinal numbers can represent an infinite number being a number that is larger than all conceivable numbers, but by using the concept of 'larger than', you are implicitly suggesting an order to the numbers.  The converse of this is the infinitesimal, which is smaller than the smallest conceivable number (quite clearly, one cannot talk about an infinitesimal sized set).

The notion that there can be different sizes of infinite sized sets is indicated by the fact that there are an infinite number of integers, yet between each integer there is an infinite number of real numbers, so the total number of real numbers must be more than the number of integers, even though both sets are of infinite size.

#### JimBob

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« Reply #6 on: 16/02/2008 19:36:09 »
This is why I never excelled at mathematics. Set theory. I personally never found a good use for it but I do not know of a practical use of it.

In Wikipedia http://en.wikipedia.org/wiki/Set_theory this is found:

Objections to set theory

"Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Errett Bishop dismissed set theory as "God's mathematics, which we should leave for God to do." Also Ludwig Wittgenstein questioned especially the handling of infinities, which concerns also ZF. Wittgenstein's views about foundations of mathematics have been criticised by Paul Bernays, and closely investigated by Crispin Wright, among others.

"The most frequent objection to set theory is the constructivist view that mathematics is loosely related to computation and that naive set theory is being formalised with the addition of noncomputational elements.

"Topos theory has been proposed as an alternative to traditional axiomatic set theory. Topos theory can be used to interpret various alternatives to set theory such as constructivism, fuzzy set theory, finite set theory, and computable set theory."

#### DoctorBeaver

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« Reply #7 on: 16/02/2008 21:17:50 »
George's post I understood. As for JimBob's - what the hell are you wittering about, boy!?

#### JimBob

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« Reply #8 on: 17/02/2008 00:38:17 »
Set Theory and how it is viewed by some mathematicians. Check out the link I quoted from. Evidently, there is no consensus about what set theory does - the quote say essentially that it is not a valid form of math (I think).

#### another_someone

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« Reply #9 on: 17/02/2008 01:23:48 »
To say there are dissenters is scarcely to say there is no consensus.

Set theory is set theory - and it works for those who find a use for it, and not for others.  Rather like complex numbers - few accountants need to use complex numbers (possibly some like to work with imaginary numbers - but that is another matter altogether), but they are useful in some branches of mathematics.

I don't claim to be very conversant with set theory, but I do recognise some of the useful answers they have provided.

Much of the objection (following the links you refer to) seem more about the casual handling of infinite sets, rather than about set theory per se.  Certainly, there are aspects of infinite sets that I would regard as counter-intuitive to say the least (such as a set that can contain two mutually exclusive subsets, yet both subsets are of the same size as the parent set).

Then, at school, I could never find a use for matrices.

#### DoctorBeaver

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« Reply #10 on: 17/02/2008 08:22:33 »
Quote
few accountants need to use complex numbers (possibly some like to work with imaginary numbers - but that is another matter altogether)

Why do "Gordon Brown" and "NHS" spring to mind?

#### Paul George

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« Reply #11 on: 24/02/2008 12:57:02 »
Hey Judge,

Did you know that they bright man that developed it - Cantor - ended his life in a mental house? he couldn't actually face the robust feelings of eternity.
He discovered that beyond infinity - Alef, the first letter in the Hebrew alphabet - there more a stronger infinity and so forth...
Can you imagine?

Paul

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« Reply #12 on: 24/02/2008 21:54:22 »
A lot of the Appollo astronauts went a bit nutty just looking out into the infinity beyond the moon.
The Master on Doctor Who went nuts looking into the time vortex. It can happen to us all. I get transfixed by those infinity mirror with the light bulbs around.

#### DoctorBeaver

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« Reply #13 on: 24/02/2008 22:05:30 »

The Master on Doctor Who went nuts looking into the time vortex.

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« Reply #14 on: 25/02/2008 18:17:54 »
Doc you can learn a lot from fiction. All I meant is that if you try too hard to comprehend something incomprehensible you may go mad or run away from the thought by busying yourself as much as you can. Back to the astronauts, you should read Moondust. It tells how the Apollo guys either thought they heard music when looking into infinity or thought they felt God or were stupidly happy or felt very alone and vulnerable.
When they returned to earth some had breakdowns, some found religion, one founded an institute to explore the unexplained and the rest worked themselves to death. Sean Bean has spent the rest of his life trying to paint how he felt on the edge of infinity.
When I was young I had a story book that had a picture of a mouse and a rabbit reading the book and on the back of their book was a mouse and a rabbit reading a book and so on and so on until the print became too tiny. It was the first thing that really freaked me out. I didn't know then that this was infinity but it really messed with my head.

#### Paul George

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« Reply #15 on: 27/02/2008 11:24:59 »
Does dealing with infinity requires to "believe"

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« Reply #16 on: 27/02/2008 17:04:25 »
Don't get your question or statement. Are we talking God here? I'm sorry but I don't believe in God but if you do, more power to your elbow. I am slightly jeleous of people that do. Some people might say that believing in a greater power is sort of running away. I'm not sure I believe that.

#### another_someone

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« Reply #17 on: 27/02/2008 18:03:06 »
Doc you can learn a lot from fiction. All I meant is that if you try too hard to comprehend something incomprehensible you may go mad or run away from the thought by busying yourself as much as you can. Back to the astronauts, you should read Moondust. It tells how the Apollo guys either thought they heard music when looking into infinity or thought they felt God or were stupidly happy or felt very alone and vulnerable.
When they returned to earth some had breakdowns, some found religion, one founded an institute to explore the unexplained and the rest worked themselves to death. Sean Bean has spent the rest of his life trying to paint how he felt on the edge of infinity.
When I was young I had a story book that had a picture of a mouse and a rabbit reading the book and on the back of their book was a mouse and a rabbit reading a book and so on and so on until the print became too tiny. It was the first thing that really freaked me out. I didn't know then that this was infinity but it really messed with my head.

Although a lot has been written about astronauts' behaviour after returning to Earth (although only about US astronauts, not cosmonauts, or the handful of other nations who have sent up (wo)men into space); but going into space is not the only thing these people have in common.  These people operate in a very intense and isolated environment, even when they are not in space, so it is understandable that when they leave that environment they can get a bit lost.

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« Reply #18 on: 27/02/2008 18:09:37 »
I guess achieving your dream in your 20's doesn't exactly leave you with any more things to conquer. Going to the moon must be hard to top. I still think that seeing the earth from space and looking out into infinity must have a strange effect on most people.

#### another_someone

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« Reply #19 on: 27/02/2008 18:35:25 »
I guess achieving your dream in your 20's doesn't exactly leave you with any more things to conquer. Going to the moon must be hard to top. I still think that seeing the earth from space and looking out into infinity must have a strange effect on most people.

I agree with all of that, but the other problem (and this was certainly the case with Yuri Gagarin, who was well short of the Moon, but the first man in space), is that it takes quite a strain to suddenly find yourself a national hero when all you want to be is a guy doing a job.

#### Relph

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« Reply #20 on: 29/02/2008 07:35:53 »
Infinity is a concept that's been used in different ways over different areas. For example you can use infinity just a placeholder for the behavior of a function that keeps increasing as you increase the independent variable(s).
But the interesting way to see infinity is found on set theory. First, you have ordinals. To get ordinals, you start from nothing, literally. You start with a set, call the empty set. Think of it as an empty bag. It's represented as {}. Now, you can get this "bag" inside another bag, let's call it {{}}. This set is now different from the first one, cause this set has something on it, the original set. You can then get these two sets into another set, to get {{},{{}}}. Again, this one is different. If you keep going, you will soon find yourself with a bunch of sets. Now, you can also order this set, by saying that a set is smaller than another if the first one is inside the other one. Because of the way we have constructed our sets, one set will either be smaller, bigger or equal to another. That's why they are call ordinals, cause they have been ordered completely.
All these sets we have made have a finite number of things inside. But what if you take all these sets and put them in another set? You will get a new set call ω. That set is basically our first infinite. But then, you could get all the initial sets, and ω, inside another set, we'll call it ω+1. You now have another infinite set!. You can keep going like this forever, and you will get infinites even larger than you could possible imagine.