[FOM] V = L mathematically

Harvey Friedman friedman at math.ohio-state.edu
Sun Feb 12 23:59:55 EST 2006

We look at order invariant relations R containedin alpha^k x alpha^k =
alpha^2k, in the usual sense. We say that R is strictly dominating iff
R(x,y) implies max(x) < max(y).

We write R# for the unique A containedin alpha^k such that

R[A] = alpha^k \ A.

THEOREM. In a weak fragment of ZF, we can prove V = L if and only if

every set of ordinals is a cross section of some R#

where R is a strictly dominating order invariant relation contained in some
alpha^k x alpha^k = alpha^2k. (Cross section obtained by fixing the first
k-1 arguments).

We can also use this as a definition of the

**constructible multivariate relations on the ordinals**

as the multivariate cross sections of the R#.

The constructible sets are then defined as the sets A such that
(TC(A),epsilon) is isomorphic to some (alpha,S), where S is a constructible
binary relation on alpha.

Harvey Friedman



More information about the FOM mailing list