Thursday, July 30, 2015

Set theory and omniscience

Grim's essay, in particular, reads like a veritable tour de force. He marshals a battery of arguments, appealing to the divine liar paradox, the paradox of the knower, Cantor's power set theorem, and essential indexicals to argue that it is impossible for there to be a known collection of literally all truths.

i) But aren't there tensions in Cantorian set theory? Even set theoretical paradoxes? So I don't see that one can safely absolutize Cantorian set theory as the standard of comparison. Any appeal will have to be selective, given the paradoxes.

ii) What about competing versions of set theory:

There are a number of different versions of set theory, each with its own rules and axioms. In order of increasingconsistency strength, several versions of set theory include Peano arithmetic (ordinary algebra), second-order arithmetic (analysis), Zermelo-Fraenkel set theory, Mahlo, weakly compact, hyper-Mahlo, ineffable, measurable, Ramsey, supercompact, huge, and Inline1.gif-huge set theory.

Does Grimm's set-theoretical objection to divine omniscience hold for all versions of set theory, or just for Cantor's?

iii) Likewise, given set-theoretical paradoxes, musn't Grimm privilege one side of the paradox to the detriment of the other? If so, on what basis? He can't apply set theory as a whole in objection to omniscience, can he? 

iv) Apropos (iii), isn't there a prima facie tension between the Cartesian product (which has no upper maxima) and the absolute infinite (which does)? 

v) Apropos (iv), doesn't modern set theory distinguish between sets and proper classes? The later is not a set (or universal set), as I understand it. For instance:

[1] On the iterative conception, the set-theoretic universe is stratified into a (well-ordered) sequence of "levels." Sets at lower levels are logically prior to sets at higher levels, and sets at higher levels depend on those sets from lower levels which serve as their members. Although the historical origins of this conception are somewhat obscure—Potter provides a nice discussion of the relevant issues in sections 3.2 and 3.9—the iterative conception has now become the standard picture for working set-theorists. Among other things, it provides a well-motivated way of avoiding the classical set-theoretic paradoxes. Since collections like "the class of all sets" or "the class of all ordinals" include sets from all levels of the hierarchy, they don't themselves form sets at any level of the hierarchy; on the iterative conception, therefore, they don't form sets at all.

vi) Most importantly, doesn't his objection crucially depend on treating truths as relevantly analogous to numbers? But since truths and numbers are disanalogous in some respects, how does he isolate the relevant commonality?

For instance, mathematical relations are necessary truths, but necessary truths are just a subset of all truths. What about contingent truths? 

Why assume that truths should be classified as mathematical sets in the first place? Isn't a numerical set a very specialized concept? Take Cantor's diagonal proof. Can you really extend that type of reasoning to a set of truths? Or is that vitiated by an equivocation, where he's using "set" in a rigorous technical sense, then applying that to a "set of truths," where it has a looser, more popular meaning? 

Many truths have a richer conceptual content than numbers. Are they really comparable? 

On the philosophical side, this section is where Potter pays the most sustained attention to the notion of dependence which underlies the iterative conception of sets. The problems with this notion are really quite severe. Although mathematicians have a well-used stock of metaphors—temporal metaphors, modal metaphors, etc.—for explaining this notion, it's not at all clear that we can cash these metaphors out into (reasonably) respectable metaphysics.

Just one of several things that should caution us against using set theory as a Procrustean bed to measure divine omniscience.

1 comment:

  1. I don't think he grasps divine omniscience. It sounds as if he's attacking a straw man.