an:03946188
Zbl 0589.06001
Winkler, Peter
The infinite random order of dimension k
EN
Publ. D??p. Math., Nouv. S??r., Univ. Claude Bernard, Lyon 2/B, 37-40 (1985).
00309821
1985
j
06A06 03C90 60C05
product order; uniform probability; random orders; first order sentence; language of ordered sets
Fix an integer \(\kappa\geq 1\) and let \(I^{\kappa}\) be the unit hypercube in Euclidean \(\kappa\)-space, endowed with the ordinary product order. If n points are chosen randomly and independently from the uniform probability distribution on \(I^{\kappa}\), the resulting ordered set is called the random order \(P_ n^{\kappa}\). The author discusses the problem, whether ''the 0-1 law'' holds for random orders, i.e. a theorem saying that for any first order sentence S in the language of ordered sets the \(\lim_{n\to \infty}P(S\) holds in \(P_ n^{\kappa})\) is either 0 or 1. The conjecture is made that the answer is yes in general, no for fixed or bounded dimension.
V.N.Salij