What if science is wrong?

[esme 10]

One thing that many scientists are wrong about, infinity, I know mathematicians and physicists hate the term.

They don't like infinity because it means they can't calculate an answer when they hit a term in an equation containing it, which means there is either something wrong with the theory they are using to make the calculation or they are missing a way of making the calculation which avoids the infinite term.

 

They understand infinity very well

 

Did you know for example there is more than one infinity ?

 

For example take the set of positive integers starting from zero and going on forever

 

This is a countably infinite set, so called because you can count each individual element, you'll never reach the end though because there isn't one.

 

Then take the set of real numbers, this is not a countably infinite set, you can't count each element one at a time, take any two real numbers which differ by as small an amount as you like and it's always possible to put another number between them, so you can't count the set, which makes that infinity larger than the earlier one.

 

Further there are even higher orders of infinity, I knew about the first two from school but the idea that there were more came as something of a surprise.

 

I found that fairly mind boggling, and I believe may have contributed to the nervous breakdowns suffered by Georg Cantor who found a way to make sense of the concept of infinity as an actual entity rather than something that could only be tended towards as a potential entity.

 

So mathematicians and physicists may not like the concept of infinity much, but they do understand it, and saying they are wrong about it just because they don't like it would be incorrect

 

Unless of course you can point to an opposing theory that explains the concept better.

[spooky3 10]

They don't like infinity because it means they can't calculate an answer when they hit a term in an equation containing it, which means there is either something wrong with the theory they are using to make the calculation or they are missing a way of making the calculation which avoids the infinite term.

 

They understand infinity very well

 

Did you know for example there is more than one infinity ?

 

For example take the set of positive integers starting from zero and going on forever

 

This is a countably infinite set, so called because you can count each individual element, you'll never reach the end though because there isn't one.

 

Then take the set of real numbers, this is not a countably infinite set, you can't count each element one at a time, take any two real numbers which differ by as small an amount as you like and it's always possible to put another number between them, so you can't count the set, which makes that infinity larger than the earlier one.

 

Further there are even higher orders of infinity, I knew about the first two from school but the idea that there were more came as something of a surprise.

 

I found that fairly mind boggling, and I believe may have contributed to the nervous breakdowns suffered by Georg Cantor who found a way to make sense of the concept of infinity as an actual entity rather than something that could only be tended towards as a potential entity.

 

So mathematicians and physicists may not like the concept of infinity much, but they do understand it, and saying they are wrong about it just because they don't like it would be incorrect

 

Unless of course you can point to an opposing theory that explains the concept better.

 

Infinity in most cases is pretty simple to understand.

 

It's when you need to prove a theorem on an infinite set that it's annoying, simply just because you can never test all possibilities, so where do you call time out... but in those cases the fact of testing the infinite set will be declared so others can take heed!

[esme 10]

Infinity in most cases is pretty simple to understand...

True, yet so many people have trouble with the idea, they just seem to think it's a big value

[dosxuk 10]

Did you know for example there is more than one infinity ?

 

For example take the set of positive integers starting from zero and going on forever

 

This is a countably infinite set, so called because you can count each individual element, you'll never reach the end though because there isn't one.

 

Then take the set of real numbers, this is not a countably infinite set, you can't count each element one at a time, take any two real numbers which differ by as small an amount as you like and it's always possible to put another number between them, so you can't count the set, which makes that infinity larger than the earlier one.

 

Except it doesn't, because of the nature of infinity!

[spooky3 10]

True, yet so many people have trouble with the idea, they just seem to think it's a big value

 

I'll double that!

[spooky3 10]

Except it doesn't, because of the nature of infinity!

 

But if you are to test it you need to decide on a value of infinity, but you can always double it (or increase it anyhow). So terms such as double infinity are used, there was a very good BBC documentary on the subject a few years ago!

 

 

Not that I think you need any help in understanding it, but:

These cases demonstrate a paradox not in the sense that they demonstrate a logical contradiction, but in the sense that they demonstrate a counter-intuitive result that is provably true

http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

[HeadingNorth 11]

Except it doesn't, because of the nature of infinity!

 

I don't think you fully understand the nature of infinity (but then, I'm not sure that anyone ever can...)

 

The set of real numbers is endless; there is no last member of it.

The set of integers is also endless, and has no last member; but it is provable that the set of integers is not as big as the set of real numbers. In other words, one endless set is more endlessly endless than the other!

 

 

I suspect that the concept would cause a lot less trouble if we used English words for it, and replaced all uses of "infinite" with "endless" and "infinity" with "endlessness." Nobody would ever argue that endless-plus-one is a bigger number than endless, because everyone knows perfectly well that endless is not a number; it's an adjective.

Edited April 19, 2012 by HeadingNorth

[Cyclone 10]

So one endless group cannot be larger than another endless group.

[HeadingNorth 11]

So one endless group cannot be larger than another endless group.

 

You'd think, wouldn't you? And yet some of them are.

 

The best explanation I've seen of it was from Asimov, who said that if you put things which are not numbers into numerical calculations, you shouldn't be surprised if you get answers which wouldn't make sense for numbers. For instance, "man + woman = trouble"

 

If you add one to, or take one away from, an endless group the group is still endless; but there are different sizes of endless group.

 

If that doesn't make sense to you, don't come back to me with it. It doesn't make sense to me either, but I've read the proofs and it definitely is true...

[spooky3 10]

But to use it you need to pick an actual number and that can always be bigger.

[dosxuk 10]

I don't think you fully understand the nature of infinity (but then, I'm not sure that anyone ever can...)

 

The set of real numbers is endless; there is no last member of it.

The set of integers is also endless, and has no last member; but it is provable that the set of integers is not as big as the set of real numbers. In other words, one endless set is more endlessly endless than the other!

 

As you say, while we can prove that a set of all integers from 0..∞ has less items than a set of all real numbers from 0..∞ (or even a set of all real numbers between 0..1), but, both sets have ∞ items in them, and while of different lengths, both lengths are equal to ∞.

[Cyclone 10]

http://en.wikipedia.org/wiki/Cardinality#Infinite_sets

 

Appears that whether they are or are not infinite sets with differing cardinality all depends on how you want to look at it.