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Surreal Numbers
Blake:poetry::Cantor:math
To see a World in a grain of sand
And a Heaven in a Wild Flower,
Hold Infinity in the Palm of your hand
And Eternity in an Hour.
Knuth on the subject:
In the beginning all was void and Conway began to create numbers.
links:
screen shots of cellular automaton using nim addition
misc:
notation: d(x) denotes the dali function applied to x, that is to say the function that maps a given real to it's analog in the surreals.
notation: bold numbers shall indicate not the number in the reals but it's analog in the surreals. ie 31337 = d(31337).
definition: call a surreal x finite if there exist y,z in R such that: d(y) x d(z). call it infinite otherwise.
proposition p1:
if a surreal x is finite, so is x + 1.
proof:
if x is finite, d(y) < x < d(z) for some y,z in R. then d(y) + 1 < x + 1 < d(z) + 1. it has been demonstrated that the mapping d is additive and multiplicative, so it follows that d(a)+b=d(a + d-1(b)). since d-1(1)=1 it is true that d(y+1) < x + 1 < d(z+1), implying x+1 is finite whenever x is, q.e.d.
proposition p2:
if a surreal x is infinite, so is x - 1.
proof:
assume not. then x is infinite but x-1 is finite, that is d(y) < x - 1 < d(z). but this means d(y)+1=d(y+1) < x < d(z)+1=d(z+1), which implies x is itself finite. this is a contradiction.
proposition p3:
the surreals are disconnected
proof: follows from definition of disconnectedness. an open set S is said to be disconnected when there are two open, nonempty subsets I and F such that intersection(I,F)={} and union(I,F)=S. let I be the infinite surreals and F be the finite ones. clearly the union is S, the intersection is empty, and the sets in question are nonempty. propositions p1 and p2 and their proofs constitute an argument that the sets are open.
Surreal Sandbox A Place to play