The definition of when two sets have the "same cardinality" is (as most people here know) when there is a bijection between the members of one set and the members of another set.

It's easy to prove beyond a shadow of a doubt that if you associate each integer N with the even integer 2N, that is a bijection between the set of integers and the set of even integers.

You may not like this definition; you may be unwilling to accept this definition (of being the same size). But you can't argue with the fact that there exists a bijection between Z and 2Z (the symbols mathematicians use for the integers and the even integers, respectively).

Someone pointed out that if you could select an integer at random (in some sense; this is very tricky to define!), then only 50% of the time would you select an even integer.

This is a valid point, but it doesn't affect the fact that Z and 2Z have the same cardinality.
What it does point out is the fact that the density of the even integers, in the integers, is only 1/2.

In case this interests you: Suppose X is a subset of the (let's say positive) integers. Let X_n mean the numbers in X that are no larger than the integer n. The fraction X_n / n tells what fraction of the first n positive integers — that is, the set {1, 2, 3, ..., n} — happens to be in the set X.

IF that fraction X_n / n approaches a limit (call it L) as n approaches infinity, THEN the set X is said have density equal to L in the positive integers. (It may be fun to try to find a set X of positive integers for which that fraction X_n / n does not approach a limit as n approaches infinity. There are many such examples.)

I disagree. Imagine (literally imagine) the dense set of odd numbers lined up in normal fashion with the natural numbers. The latter is greater. You can put 3 next to 2 from the odd the the naturals, and do that to infinity. But how does this not distort the infinity of the odd numbers? We already said it has half the density

What do you disagree with?

Sorry for the intrusion. Just checking a link and now can't get rid of it! See my new thread. :sad:

Doubling the density of the odd numbers in order to get a cardinality equal to the naturals.

Density and cardinality are different concepts.

Density and cardinality are different concepts. — Daz

I think density is the true cardinality, and when you double the density in order to get the cardinality of the odd to equal that of the naturals you've made an invalid move

Is this what you are talking about? :

https://www.encyclopediaofmath.org/index.php/Density_of_a_set

That's the general idea, Dr. G., but to be exact I'm thinking of https://en.wikipedia.org/wiki/Natural_density .

OK, thanks. Another concept I was unfamiliar with! :smile:

I've yet to see anyone justify the maneuver of moving the odd numbers over to "line up" with the naturals.

I don't "know" what you "mean".

It's not that difficult