Are there more even numbers than odd numbers?
Very simple 'yes-or-no' question, but I can't find the answer anywhere. My gut feeling says the number of odd and even numbers are equal but I managed to write up something that contradicts my intuition. Although I still think my gut feeling is right, I can't find any logical or mathematical errors with my "proof". Can somebody please look it over and tell me which one of me is right?
Statement 1: For every positive odd integer $o$ there is an even integer $e=o+1$.
Statement 2: For every positive integer $n$ there is an negative integer i.e. $-n$.
Conclusion 1: The number of positive odd integers ($O_{positive}$) is equal to the number of positive even integers ($E_{positive}$). If there is a negative equivalent for every positive integer then the number of negative odd integers ($O_{negative}$) is equal to the number of negative even integers ($E_{negative}$). In short: $$O_{positive}=E_{positive}=O_{negative}=E_{negative}$$
Statement 3: The number zero is "neutral" (neither positive nor negative).
Statement 4: The number zero is an even integer.
Conclusion 2:
$$\begin{align} O_{total} & = O_{positive} + O_{negative} + O_{neutral} \\ & = O_{positive} + O_{negative} + 0 \\ & = O_{positive} + O_{negative} \end{align}$$
And:
$$\begin{align} E_{total} & = E_{positive} + E_{negative} + E_{neutral} \\ & = E_{positive} + E_{negative} + 1 \\ \end{align}$$
So:
$$\begin{align} E_{total} & = O_{total} + 1 \\ E_{total} & > O_{total}\\ \end{align}$$
Right? As an engineer I use math daily, but that doesn't make me a mathematician. So please be gentle :)
First things first - there are an infinite number of both even numbers and odd numbers.
It's important to realize that $\infty$ (infinity) is not a number. Therefore it doesn't really make sense to talk about the "number" of even or odd numbers, or to write statements like $E_{\rm even}+1$, because that's assuming that $E_{\rm even}$ is a number that you can sensibly add $1$ to.
However, perhaps surprisingly it does make sense to ask if there are more even numbers than odd numbers. That is, you can compare two infinite quantities, or compare a finite quantity and an infinite quantity, even if you can't meaningfully add and subtract infinite quantities.
They way we define more, less and the same for infinite quantities is as follows. For two collections $A$ and $B$ (say $A$ are the even numbers and $B$ are the odd numbers) we say that
If you can associate every item in $A$ with a unique item in $B$, and vice versa, then $A$ and $B$ are the same size.
If you can associate every item in $A$ with a unique item in $B$, but not vice versa, then $B$ is bigger than $A$.
If you can associate every item in $B$ with a unique item in $A$, but not vice versa, then $A$ is bigger than $B$.
In your case, you can associate every even number $n$ with the odd number $n+1$, and you can associate every odd number $m$ with the even number $m-1$ (assuming 0 is even) so therefore there are just as many odd numbers as even numbers.
This can lead to seemingly paradoxical results, because e.g. you can associate every whole number $n$ with the even number $2n$, and every even number $m$ with the whole number $m/2$, so there are just as many even numbers as whole numbers, even though the even numbers are a subset of the whole numbers.
As there is a bijection $$ f(x) = x + 1 $$ sending any odd number to an even, this shows that the sets have equal size.
Here, I assumed that the natural numbers start with $1$, if they should start with $0$, simply define the same function on the even numbers.