FOM: arbitrary and random

[Kanovei]

Date: Wed, 30 Jan 2002 17:19:13 -0500
From: Harvey Friedman <friedman at math.ohio-state.edu>

>So now let us assume that "arbitrary" is a predicate on objects. Let us see
what axioms we want to place on this concept.

There are studies of extensions of ZFC by the predicate "random" 
which follow approximately these ideas, see M. van Lambalgen JSL, 
1992, where also conservativity results are discussed. 

>2. (forall x)(Arb(x) implies phi) or (forall x)(Arb(x) implies notphi),
where phi is any formula of first order predicate calculus with equality
not mentioning Arb, all of whose free variables are x.

But this seems to be too strong a requirement, indeed, take 
phi to be ODD, shall we claim that an arbitrary natural number 
is necessarily odd or necessarily even ? 
For real numbers take x>0 as phi. 

V.Kanovei