What alternative measures of infinite sets exist?

Cardinality measures the set of even numbers to be equal in size to the set of natural numbers, which I have no problem with. Are there accepted alternative measures of size which do recognise the fact that there are two natural numbers for every even number?

For example, the even numbers and the odd numbers are two disjoint subsets equal in cardinality which can be placed in union to form the natural numbers. The direct union of the two distinct subsets into the third is surely a stronger measure of magnitude than cardinality since it preserves the index of the elements.

If we take the set of all natural numbers and remove the element $1$ from it, we can compare this new set with the natural numbers and say that the larger set contains every element which the smaller contains plus one further element which the smaller set does not. Cardinals don't allow us to say that this set is one larger. Is there some system that does?

• You are probably looking for density. – Simply Beautiful Art Jan 18 '17 at 21:36

1 Answer

Measure theory offers another kind of formalizing the intuitive concept of size.

Let, for example $\{a_n\}$ is sequence of positive numbers for which

$$\sum_{n=1}^{\infty}a_n<\infty.$$

Then, let's the measure of the size of a set of positive integers $\{i_1,i_2,\cdots\}$ by

$$\sum_{n\in\{i_1,i_2,\cdots\}}a_n.$$

For the set of positive odd numbers we have now

$$\sum_{n=0}^{\infty}a_{2n+1}$$

and for the positive even number we have

$$\sum_{n=1}^{\infty}a_{2n}.$$

• It looks like projecting the number line down to the segment from 0 to 1, would be much the same as this, and then measuring what proportion of the segment is included. Do I understand correctly? – samerivertwice Jan 19 '17 at 8:27
• If we say $a_n=\frac{ 1}{ 2^ n}$ for example then we can measure these? But the even numbers greater than zero will measure smaller than the odds by this measure. Is that correct? – samerivertwice Jan 19 '17 at 8:34
• Yes. Those sizes will be different. – zoli Jan 19 '17 at 9:54