Following on from the discussion of the set of all the integers, with order , and the set of all its subsets– the continuum, of order with – what can we say about the set of all subsets of the continuum?
Like any other set of order we can say it has order , in this case or . But what do its members represent in the physical world?
By interleaving digits each point on the continuum can also be thought of as a representation of cartesian co-ordinates: for instance 0.55011200…. can be thought of as (0.5010…, 0.5120…) using interleaving, so we can think of a subset of the continuum as a subset of points on the cartesian plane and all the subsets as all the possible curves in the cartesian plane (including disjoint curves).
According to Wheels, Life and Other Mathematical Amusements– the book is a good thirty years old so I am hedging it here a bit – there are no known physical representations of yet higher transfinite orders – they exist in a purely mathematical world: discussion of the philosophical implications of which is fascinating but beyond me.