Something odd about the number of odd numbers

[SecularCauses]

Looking at the natural, or counting, numbers, we have an infinite set of numbers alternating between odd and even numbers, {1,2,3,4,5,6....}. Seeing the odd and even numbers side by side like this, we would reasonably conclude that there are as many odd numbers as there are even numbers. This conclusion, however, is false.

If we could only count to three, and we had to solve the problem of whether the right and left hand of a person had the same number of fingers, one may think that the problem would be unsolvable. How can one figure out that two hands, each with five fingers, have the same number, when one can only count to 3? If counting were the most fundamental math activity, then this would probably be an unsolvable problem for those who cannot count up to 5. However, matching is even more fundamental than counting. The person who can only count to 3 can solve the problem by simply asking if there is a way to match every finger on the right hand with every finger on the left. Since one can match each finger on the right hand to a finger on the left, even if one does not know the number of fingers existing on each hand, one may say that they are equal in number.

We are like the person who can only count to 3 when dealing with the problem of trying to count all the odd numbers or all of the even numbers. There is no way we can ever get to the end and count them all. So, if we are asked whether there are more even numbers than odd numbers, we have to use a matching principle to solve the problem. When we do, we discover that the total number of even numbers equals all of the even numbers combined with all of the odd numbers combined. The matching principle is easily seen below:

1 -------> 2
2 -------> 4
3 -------> 6
4 -------> 8

n -------> 2n

In other words, for every odd number, if we multiply it by 2, we get a unique even number we can match to the odd number. And, for every even number, we can multiply it by 2 as well, to get a unique even number to match to the even number. Therefore, the even numbers equal the even numbers plus the odd numbers.

Of course, if one could match all odd numbers to all odd and even numbers combined, the above conclusion would fail. There is, however, no such matching principle.

So, although odd numbers sit side by side with the even numbers, there is something odd about them. They are far less than the even numbers.

So, how are there so many more even numbers than odd numbers when they alternate between each other out to infinity? What is it about being an odd number that leads to an infinite number of odd numbers being smaller than an infinite number of even numbers?

[mickthinks]

Of course, if one could match all odd numbers to all odd and even numbers combined, the above conclusion would fail. There is, however, no such matching principle.

The function f(n) = 2n - 1 maps the set of odd and even numbers onto the set of odd numbers ...

1 -------> 1
2 -------> 3
3 -------> 5
4 -------> 7

n -------> 2n - 1

[SecularCauses]

Of course, if one could match all odd numbers to all odd and even numbers combined, the above conclusion would fail. There is, however, no such matching principle.

The function f(n) = 2n - 1 maps the set of odd and even numbers onto the set of odd numbers ...

1 -------> 1
2 -------> 3
3 -------> 5
4 -------> 7

n -------> 2n - 1

About time someone figured it out.

So, how can mathematicians claim that there are as many even numbers as there are even and odd numbers combined, when we can do the same matching trick for odd numbers, and conclude that there are as many odd numbers as there are odd numbers and even numbers combined? Did Cantor make a mistake?

[Impenitent]

there may be an equal amount of non zero, odd and even numbers...

doesn't counting zero as even give one more to the evens?

-Imp

[Mike Strand]

This might help:

In math, “counting” is generalized to “one-to-one, onto matching”. Thus counting four objects can be accomplished by matching each member of the set {1,2,3,4} to one and only one of the objects.

“Size” is generalized to “cardinality”.

The key is to define what it means to say that a collection or set is finite (consisting of a finite number of objects). A set is finite if there is a natural number N so that the set has the same cardinality as the set {1,2,3,…,N}, which is the set of all natural (counting) numbers up to a largest but finite number N. That is, a finite set can be matched one-to-one, and onto the set {1,2,3,...,N} for some natural number N.

This allows “infinite set” to be defined simply as a set that is not finite. This means that an infinite set can't be covered by a one-to-one matching with the set {1,2,3,...,N}, no matter how large N is chosen to be. This is a simple definition, but with weird implications.

The set or collection of all natural numbers can be written as {1,2,3,...,N,N+1,...}. To show that it is infinite, consider any natural number L, where L is as large as I want it to be, and I try to cover the set of all natural numbers with {1,2,3,...,L} with a one-to-one matching. I find that L+1, L+2, and so on, are natural numbers that haven't been covered. No matter how large L is, there are still larger natural numbers. Thus the set of all natural numbers is not finite -- i.e., it is infinite.

The set of all natural numbers, the set of all odd numbers, and the set of all even numbers have the same cardinality, which is the cardinality of the set of all natural numbers, {1,2,3,…,N,N+1,…}. That is, these three sets all have the same “size”, in a manner of speaking. Matching N to 2N, and N to 2N-1 shows this. Thus we have two infinite sets, the odds and the evens, that have the same “size” or “cardinality” as the set of all natural numbers, but they are both subsets of the natural numbers!

This odd, unintuitive state of affairs only happens with infinite sets, and it is a reason why “infinity” itself is difficult to grasp directly.

For another example, it can be shown that the set of all rational numbers -- fractions of the form M/N, where M and N are integers (positive or negative) -- also has the same cardinality as the set of all natural numbers -- which itself is a subset of the rational numbers!

By the way, there are cardinalities that exceed that of the set of all natural numbers. This means that there are sets that cannot be covered by a one-to-one matching with the set of all natural numbers. That is, there are different degrees of infinity. That's the basis of another topic, maybe.

[Mike Strand]

To Impenitent: The set {0,1,2,3,..., N,N+1,...} has the same cardinality (size) as the set {1,2,3,...,N,N+1,...,}, the set of all natural numbers. To show this, match N from the second set with N-1 from the first set, for each natural number N. This is a one-to-one and onto matching.

It can also be shown that the set of all (positive and negative) integers including zero, {..., -N-1,-N,...,-3,-2,-1,0,1,2,3,...,N,N+1,...} has the same cardinality as the set of all natural numbers (positive integers). Match 1 with 0, 2 with 1, 3 with -1, 4 with 2, 5 with -2, and so on. That is, match N with -(N-1)/2 if N is odd, and match N with N/2 if N is even, for each natural number N. This is a one-to-one and onto matching.

[Mike Strand]

To SecularCauses, mickthinks, and Impenitent:

Did my comments help at all? Were they clear and believable, or did they just fog things up? Or maybe just boring as hell? Any comment on my examples?

Thanks, SecularCauses, for what I think is an interesting and challenging topic.

[Impenitent]

I understand how the infinite sets match up infinity=infinity+1, it just doesn't feel right

-Imp

[Mike Strand]

Thanks for your comment, Impenitent.

"Infinite" is more different from "finite" than I first expected, too. Not exactly intuitive, even though the logic is clear. It's almost like saying that "infinite" is so much bigger than "finite", or one, that adding a finite number of elements, or one element, to a set made up of an infinite number of elements doesn't change its cardinality (size).

Even more puzzling is that the combination of all the negative integers, zero, and all the positive integers still has the same cardinality as the set of all positive integers, using the one-to-one matching logic that we use to count finite sets.

It's interesting that the logic used to count finite sets, when applied to infinite sets, gives such unexpected results.

Anybody interested in looking at sets which have cardinality greater than that of the set of all positive integers? In other words, there are "larger infinities" than the one I've been showing here. Probably best to refer readers to any number of good text books on set theory/cardinality, or on real analysis -- or even articles in Wikipedia. This is all based on the idea I use in everyday counting: One-to-one, onto matching of elements between the "counting set" and the "set being counted".

Maybe that's the problem -- trying to "count" the elements in an infinite set in the same way I count the elements in a finite set. But how else? Is there any other way to determine, or maybe rather to define, the "size" of a collection of objects? Good question, I think.

I've read that there are cultures in which one, two, and many are the only three "sizes". This counting system would define the size of every set I've been showing here!