  Paul Erd\H{o}s and Alfred Renyi considered the evolution of the random graph
G(n,p) as p ``evolved'' from 0 to 1. At p=1/n a sudden and dramatic change
takes place in G. When p=c/n with c<1 the random G consists of small
components, the largest of size Theta(log n). But by p=c/n with c>1 many of the
components have ``congealed'' into a ``giant component'' of size Theta (n).
Erd\H{o}s and Renyi called this the double jump, the terms phase transition
(from the analogy to percolation) and Big Bang have also been proferred. Now
imagine an observer who can only see G through a logical fog. He may refer to
graph theoretic properties A within a limited logical language. Will he be able
to detect the double jump? The answer depends on the strength of the language.
Our rough answer to this rough question is: the double jump is not detectible
in the First Order Theory of Graphs but it is detectible in the Second Order
Monadic Theory of Graphs.
