Sets Have Simple Members
arXiv:1107.1458
Abstract
The combined Universal Probability M(D) of strings x in sets D is close to max M({x}) over x in D: their ~logs differ by at most D's information j=I(D:H) about the halting sequence H. Thus if all x have complexity K(x) >k, D carries >i bits of information on each its x where i+j ~ k. Note that there are no ways to generate D with significant I(D:H).
This paper has been withdrawn by the authors. Withdrawn (v8) by S. Epstein. Reposted as arXiv:1403.4539 by the coauthor