Thread: Countable and Uncountable Infinities

Hey,

I'd just like to take a 'debate' I was involved in on a previous forum regarding the nature of infinity.

People were debating (heatedly) whether there are more numbers between 0 and 1 than 1 and 2.

I invoked Cantor, and his concept of 'countability'. The set of all real numbers is uncountable, therefore it is a 'transfinity', which may or may not be smaller or larger than an infinity, but cannot be counted in such a way.

So, my position was that asking whether there's more numbers between 0 and 1 than 1 and 2 is meaningless since we cannot count them at all so no comparison can be made.

I just want a sanity check to make sure I'm not goofing up here - anybody got anything to add to that (or criticisms as to why I've completely got this wrong or something?!)

Thanks!

Whoops: Just read the infinities equations thread.. you guys are cool man, you've discussed things I find interesting even before I joined!
b.t.w. I think William is pretty much putting across the point I'm mentioned here, although in somewhat better detail. Three cheers for William!!

Infinity is a concept it is not a number. you can go on adding to 1.999999999999 till the end of time,your life,the keyboard, boredom sets in, or you use up all the memory in your PC. It can always be added to. A common mistake often made is when a person attempts to 'add' one to infinity, since infinity is NOT a number such an addition is not a mathematical proposition. Our brains are not programmed to comprehend infinity so it is a rare occasion when it is best to just accept it. A second mistake is to compare infinity to a number.


Welcome to the forum of infinite wisdom. 8)

Originally Posted by billco

Infinity is a concept it is not a number. you can go on adding to 1.999999999999 till the end of time,your life,the keyboard, boredom sets in, or you use up all the memory in your PC. It can always be added to. A common mistake often made is when a person attempts to 'add' one to infinity, since infinity is NOT a number such an addition is not a mathematical proposition. Our brains are not programmed to comprehend infinity so it is a rare occasion when it is best to just accept it. A second mistake is to compare infinity to a number.


Welcome to the forum of infinite wisdom. 8)

Yes, I said that on the forum we were having the 'debate' in, it is indeed a concept, but in a sense one can count infinities with cardinal aleph null, but not ones with cardinal aleph one.

I think the fact is, stating 'how many numbers between 0 and 1' is nonsensical because it's intuitively not countable given that whatever range you look at you get an infinity, whereas if you had a set of countable numbers, any range except the entire set is a finite number. Another point I made :-)

I hope I'm not missing things here horribly..

Oh, and thanks for the welcome :-) I'm honoured!!

Originally Posted by miscast

I just want a sanity check to make

Silly me, I singularly failed to understand your point precisely. More numbers between 1 and 2 than numbers between 0 and 1

NO - It's linear. Just add 1 to the lower set or subtract 1 from the upper set, you will create an equally valid number.

Originally Posted by billco

Originally Posted by miscast

I just want a sanity check to make

Silly me, I singularly failed to understand your point precisely. More numbers between 1 and 2 than numbers between 0 and 1

NO - It's linear. Just add 1 to the lower set or subtract 1 from the upper set, you will create an equally valid number.

Ah, but surely we can't count it anyway? So the comparison is invalid before we even get to that?

If we are going into cardinality questions you have to be very careful of your definitions. Two sets have the same cardinality if you can find a bijection between them. Bilco offered an easy one, so [0, 1] and [1, 2] have the same cardinality - in fact both have the same cardinality as the reals so in the strict mathematical sense of "having the same number of elements" they do.

However, they have more elements then the rationals or integers (as any power set has strictly more elements then the base set, proof left as an exercise for the reader - lol)

At 70 I'm way past arguing the finer points of maths (many of which have faded or blurred long ago) - I see it as common sense: Any number you can think of between 1 and 2, I can add one onto it!. - Think negative numbers if you like. I agree the range of numbers between 1 and 2 is in theory unlimited. In practice it is always limited to the accuracy required.

Though its more interesting that (0, 1) ahs as many numbers as the real line itself (even though it is contained in the real line) - this property forms the definition of an infinite set

sort of (Infinity^Infinity)^2/5 etc lol

Originally Posted by miscast

So, my position was that asking whether there's more numbers between 0 and 1 than 1 and 2 is meaningless since we cannot count them at all so no comparison can be made.

"Uncountable" does not mean "cannot be compared", it just means there is no bijection between (0,1) and the naturals. It makes perfect sense to ask whether there are more numbers between (0,1) or (1,2), or any two sets you like.

There are 'more' elements in a set B than set A if you can find an injection (a 1-1 map) from A to B but NO bijection exists. No bijection exists means they have different cardinality, while the injection means we can 'fit' A into B. This is obviously not the case for (0,1) and (1,2) which have a bijection between them and therefore the 'same number of elements'.

Originally Posted by shmoe

Originally Posted by miscast

So, my position was that asking whether there's more numbers between 0 and 1 than 1 and 2 is meaningless since we cannot count them at all so no comparison can be made.

"Uncountable" does not mean "cannot be compared", it just means there is no bijection between (0,1) and the naturals. It makes perfect sense to ask whether there are more numbers between (0,1) or (1,2), or any two sets you like.

There are 'more' elements in a set B than set A if you can find an injection (a 1-1 map) from A to B but NO bijection exists. No bijection exists means they have different cardinality, while the injection means we can 'fit' A into B. This is obviously not the case for (0,1) and (1,2) which have a bijection between them and therefore the 'same number of elements'.

Thank you :-) I was wrong then, but I was barking up the right tree I guess.

I guess then it wouldn't matter whether you asked whether there were more rational/natural/etc. numbers between in 1-2 than 0-1 or more real numbers in 0-1 than 1-2. As long as they have the same cardinality then there is an equal number in each set.

Still, I think if we talk about any infinite set where any sub-set is also infinite then comparison between sub-sets is pointless since they'll always have the same cardinality.

If we looked at two subsets of the integers then of course they're finite and can be compared.

My point was that talking about them as 'countable' in the same way that you'd compare two finite sets is wrong.

Thanks for all your advice guys! This is really a fascinating subject.

Oh and one more thing, though I think I get the thrust of your argument shmoe, what is a bijection?

Originally Posted by miscast

Oh and one more thing, though I think I get the thrust of your argument shmoe, what is a bijection?

A bijection is a 1-1 and onto mapping between sets. A function f:A->B is

1-1 or an injection: if f(x)=f(y) implies x=y. No two elements in A go to the same element in B.

onto or a surjection: if for every z in B we can find an x in A where f(x)=z. Every element in B get's mapped to be something in A.

Originally Posted by miscast

Still, I think if we talk about any infinite set where any sub-set is also infinite then comparison between sub-sets is pointless since they'll always have the same cardinality.

No it's not pointless. Not every subset of an infinite set has the same cardinality as the entire set, the rationals are a subset of the reals, {1,2,3} is a subset of the reals, and so on. In an infinite set, the entire set has the same cardinality with *some* proper subset, not all of them.

Originally Posted by miscast

If we looked at two subsets of the integers then of course they're finite and can be compared.

My point was that talking about them as 'countable' in the same way that you'd compare two finite sets is wrong.

It's not though, it's a direct analogue to how we compare finite sets. You compare finite sets with bijections, how many elements in {a, g, t, e}? Counting them a-1, g-2, t-3, e-4 is equivalent to finding a bijection to {1,2,3,4}, so we declare {a, g, t, e} has 4 elements. There's no bijection between {1,2,3} and {a,g,t,e}, but we find a 1-1 map from {1,2,3} to {a,g,t,e} so we would be justified in saying the latter has more elements than the former. This is what's going on when comparing infinite sets as well.

Originally Posted by miscast

Whoops: Just read the infinities equations thread.. you guys are cool man, you've discussed things I find interesting even before I joined!
b.t.w. I think William is pretty much putting across the point I'm mentioned here, although in somewhat better detail. Three cheers for William!!

Thank you very much.


Cheers,
william