| omniscience_cantorian.pdf |
4.4 Omniscience and Cardinality: Patrick Grim (1988) has objected to the possibility of omniscience on the basis of an argument that concludes that there is no set of all truths. The argument (by reductio) that there is no set T of all truths goes by way of Cantor's Theorem. Suppose there were such a set. Then consider its power set, ℘(T), that is, the set of all subsets of T. Now take some some truth t1. For each member of ℘(T), either t1 is a member of that set or it is not. There will thus correspond to each member of ℘(T) a further truth, specifying whether t1 is or is not a member of that set. Accordingly, there are at least as many truths as there are members of ℘(T). But Cantor's Theorem tells us that there must be more members of ℘(T) than there are of T. So T is not the set of all truths, after all. The assumption that it is leads to the conclusion that it is not. Now Grim thinks that this is a problem for omniscience because he thinks that a being could know all truths only if there were a set of all truths. In reply, Plantinga (Grim and Plantinga, 1993) holds that knowledge of all truths does not require the existence of a set of all truths. He notes that a parallel argument shows that there is no set of all propositions, yet it is intelligible to say, for example, that every proposition is either true or false. A more technical reply in terms of levels of sets has been given by Simmons (1993), but it goes beyond the scope of this entry. See also (Wainwright 2010, 50–51).
--Summary of the debate taken from SEP entry on Omniscience
--Summary of the debate taken from SEP entry on Omniscience
I think another solution can be found here:
http://www.reasonablefaith.org/does-god-know-an-actually-infinite-number-of-things
http://www.reasonablefaith.org/does-god-know-an-actually-infinite-number-of-things
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