How about if one of those infinities is not linear, without any kind of order, but random? Why not?
We have a naive and simple idea of the infinite. In the best of the cases, we think of the infinite as the unending series 1, 2, 3, ... In the worst scenario, we think of the infinite as the quantity of the grains of sand in all the deserts and beaches on the Earth.
It was Archimedes of Syracuse, more than two millennia into the past, who proved that it was impossible that the grains of sand were infinite because he was able to give a fair estimate of the grains of sand that could be placed in a sphere the size of the orbit of Saturn.
He knew that it was impossible to count grain by grain all the deserts and beaches. So it is impossible to prove that the grains of sand are finite by enumerating them one by one.
So, his approach was to establish an upper limit to the number of grains that can hold the planet Earth. This is an indirect proof that the sand cannot be infinite.
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| The Sand Reckoner. |
But what if we have an infinite book in our hands, how big can it be? Infinite in size? Infinite in weight?. Infinite in volume?
Have you ever heard of The Book of Sand? It’s a short story about an infinite book with no beginning and no end. But this book has a finite amount of pages; how come?
The infinite is incredible and surprising! What is going to limit the limitless?
Read the article Three unexpected behaviors of the infinite and see three unforeseen aspects of the infinite.