Counting to Infinity and Beyond

When you count a set of objects you are in effect labelling them:

Let's use this second way of labelling when we try to count an infinite number of objects:

The number of objects is given by the next label to be used. This label cannot be any of our 'nomal' finite numbers. The label used is the greek letter omega: . So we have counted objects so far. Now we have a surprise in store:

But try counting the objects the other way round:

Since the last label used is then you have counted to +1 which is the next label to be used. So counting beyond our normal finite numbers into the transfinite numbers springs an amazing surprise:

You might like to figure out the meaning of 2 x and x 2 . Are they the same?