## Empty one year ago What's wrong with this "proof" that there are more even numbers than odd numbers? Take all the odd numbers. Now multiply each one by 2. We now have the same number of odd numbers and even numbers. However we have left out numbers like 4 and 12. So it appears that there are more even numbers than odd! =P

1. anonymous

Hmmm, well I suppose one issue is that $\infty< \infty+ 2$or even: $\infty < 2\times \infty$Isn't quite valid when talking about cardinality.

2. Empty

Hmm what does cardinality mean exactly? I thought I was showing a 1 to 1 correspondence between the odds and a subset of the evens. But I don't really know what I'm doing.

3. perl

IF we take all the odd numbers. 1,3,5,7,... multiply by 2 2,6,10, 14,... it appears by correspondance the number of odd numbers is the same as a subset of even numbers

4. anonymous

Cardinality is going to be a natural number for finite sets, countably infinite (e.g. natural numbers), or not countably infinite (e.g. real numbers).

5. Empty

Yeah exactly what I'm thinking @perl. But I guess I also have some base assumption that there are just as many even as there are odd numbers, but maybe even this is just as invalid of a statement to be making in the first place...? In what sense can I concretely say what my intuition is thinking?

6. Empty

I guess I think the even numbers are "countably infinitely larger" than the odd numbers since we have: 1,3,5, 7, 9, ... but here I can list an infinite amount: 2, 6, 10, 14, 18,... 4, 12, 20, 28, 36,... 8, 24, 40, 56, 72,... ....

7. anonymous

Consider the following: Map every natural number to 5 times itself, which is the subset of all natural numbers divisible by 5, such is a one to one correspondence. This doesn't mean there are more natural numbers than natural numbers.

8. anonymous

That's because, in terms of cardinality, we say $$\infty \times 5 = \infty$$.

9. Empty

So is "cardinality" the same as "amount" of something? I feel like there's some sort of difference between the two. Like for instance there are "more" natural numbers than there are squares of natural numbers, but their cardinality is the same. I think there's a difference worth making here and something else that's not cardinality.

10. anonymous

Or, to be more specific $$\aleph_0 \times 5 = \aleph_0$$.

11. anonymous

Cardinality is a property of sets that can be understood as the size of the set.

12. Empty

Hmmm so are infinite sets just fundamentally broken? The fact that $$\sum_{n=1}^\infty \frac{1}{n}$$ diverges but $$\sum_{n=1}^\infty \frac{1}{n^2}$$ converges just makes me uneasy saying that natural numbers and square of natural numbers have the same cardinality and then say "that's the end of it" since it just doesn't feel like cardinality is precise enough.

13. perl

let me edit this. If we take all the odd numbers. 1,3,5,7,... multiply by 2 2,6,10, 14,... it appears there is a one to one correspondance between the number of odd numbers and a *proper* subset of the even numbers. So one might think there are in fact more even numbers than odd numbers. But this is not a valid conclusion because an infinite set can be equivalent to a proper subset of itself. The classic example is the natural numbers N is equivalent to the even numbers , which is a proper subset of N, by the function rule: n -> 2*n Hence the natural numbers are equivalent to a proper subset of itself. We can even find a bijective function from the the set of even numbers : 2,4,6,8, .. to the set : 2, 6, 10, 14 and since we already have a correspondance between the set of odd numbers and the even number proper subset 2,6,10,14 that proves that the the set of odd numbers are equivalent in size to the even numbers. There are easier ways to draw this conclusion of course.

14. anonymous

Both $$\{1/n|n\in \mathbb N\}$$ and $$\{1/n^2|n\in \mathbb N\}$$ have same cardinality as $$\mathbb N$$.

15. anonymous

The sum of the elements in a set have no relation to cardinality.

16. Empty

So what's the concept I'm looking for if it's not cardinality?

17. anonymous

No, cardinality is the concept you're looking for when it concerns the whole even/odd numbers

18. anonymous

It's just that finding a one to one correspondence between infinite sets doesn't really show anything.

19. Empty

Ok, the even and odd thing is just a bit of nonsense that is cleared up to me now. However I still feel like there is some conception of the squares of natural numbers being smaller than the natural numbers that isn't cardinality where I can say, "Yeah in this sense the natural numbers is a larger set than the squares of natural numbers".

20. anonymous

For example, take all even numbers, add 1, then you get all the odd numbers excluding 1. This means there are more odd than even numbers by your logic.

21. anonymous

You might better find an error in your proof if you formalize it more

22. Empty

Nah I agree that it's false because of the fact I'm using an infinite set and any infinite set can be shown to be in correspondence with any other infinite set. I guess I'm just not satisfied that this is the state of things, it just seems so lousy and broken haha.

23. anonymous

Yeah, in my opinion it is due to infinity not obeying most rules of algebra, particularly: $x < x+1$