The even number paradox

Morning all. Remember when you were younger and you'd play the "What is the biggest number you can think of game? Eventually some smart Alec would get to infinity plus one (which isn't actually possible but that is a post for another day) but what smart Alec is doing reflects the fact the natural numbers can go on for ever 1,2,3,4,5,6.......to whatever then plus +1. So there are literally an infinite number of natural numbers.

So it goes without saying that there are more natural numbers than even numbers, after all, the natural number set contains all the even and odd numbers. Right? Wrong! The set of natural numbers is no larger than the set of even numbers. At this point you're probably thinking this is some sort of clever trick with numbers to create some fancy paradox but its actually fairly easy to explain (After all I am writing this)

If you think it through, every natural number has a number twice as large as it and it follows that every even number has a natural number half its size. For example

1------------2
2------------4
3------------6
4------------8
5-----------10
6-----------12

And on and on. What we have done is set up a natural correspondence between the natural numbers and even numbers. It clearly follows that as we go up the number line every number will have a corresponding even number hence the natural number set will never be larger than the even number set. That is, they are both countably infinite 

So, there we have it, there are as many even numbers as natural numbers.

Maths, eh? Remarkable. Anyhoo, on a lighter note......

A Mathematician, a Biologist and a Physicist are sitting in a street cafe watching people going in and coming out of the house on the other side of the street. First they see two people going into the house. 

Time passes. After a while they notice three persons coming out of the house.
The Physicist says: "The measurement wasn't accurate.".
The Biologist says: "They have reproduced".
The Mathematician muses: "If exactly one person enters the house then it will be empty again."

Until, next time. Tatty bye.