Grim's Cantorian Argument Against Omniscience
Philosopher Patrick Grim since the early 1980s has advanced an argument against the existence of God which turns on an incoherence he sees in the concept of omniscience, an essential property of God as understood by most orthodox incarnations of the Abrahamic religions. [ 1 ] It runs as follows: There does not exist a set T of all distinct truths. For suppose towards a contradiction that T exists. Let f be a mapping from T into the set P(T) of all subsets of T, and consider the subset S of T of every truth q which is not contained in the set f(q). By definition of S, no truth q is mapped by f to S, and we conclude that f is not surjective. Since f is an arbitrary mapping from T into P(T), it follows that no such map is surjective. Now define a map g from P(T) into T. For each subset A of T (where A is a member of the set P(T)), define g(A) by the truth expressed by the sentence "A is a subset of T." Since every member of P(T) is distinct, then g maps them to distinc