Monday, December 13, 2010

Cantor’s Paradox again

This is the fifteenth of 17 posts, which are collectively Eternity, etc.
......Because of all those unions (see previous post), our collection is a nested hierarchy of sets, whose cardinalities the Beths are defined to be. And so because our collection is not quasi-spatial, nor are the Beths, collectively. So even on the modern view, cardinal arithmetic is indefinitely extensible [i]. And while that result is of a kind with Cantor’s Paradox [ii], it is the belief that cardinal arithmetic is timeless that makes it paradoxical. The atemporalist faces some tough choices, because the truths known by a timeless God would be collectively quasi-spatial, rather than variable.
......But an everlasting God could acquire arithmetical knowledge endlessly. And Presentist time is merely our natural reification of the possibility of change, not a real dimension. And under Presentist Open Theism, that possibility originates with the greatest conceivable being’s power to change. So such a God would have enough time to know each arithmetical truth, and to know it arbitrarily quickly. Time would then be indefinitely extensible, and absolutely continuous [iii], in the sense that for any duration, and for any Actual Infinite cardinal number (that could ever exist), that duration has more than that many instants (possible instantaneous changes).
......It seems, then, that only a God with the power to change is, for every Actual Infinite cardinal number, able to know all about a possible world of so many things, and hence able to create such a world perfectly freely (with a perfect understanding of His options). So the argument at the end of section VI becomes an argument that God is, since omnipotent, not timeless. And note that the informality of this rather mathematical section does not make that a weak argument. Formal proofs can only prove theorems within axiomatic systems, and since the justification of such axioms is necessarily informal, so informality also suits a more direct argument about metaphysically possible creations.
......Notes:
......[i] For more details, see W. D. Hart, “The Potential Infinite,” Proceedings of the Aristotelian Society 76 (1976): 247–64; Alvin Plantinga & Patrick Grim, “Truth, Omniscience, and Cantorian Arguments: An exchange,” Philosophical Studies 71 (1993): 267–306; Stewart Shapiro & Crispin Wright, “All Things Indefinitely Extensible,” in Agustin Rayo & Gabriel Uzquiano (eds.), Absolute Generality (Clarendon Press, 2006), pp. 255–304; Nicholas Rescher & Patrick Grim, “Plenum Theory,” Noûs 42 (2008): 422–39.
......[ii] For Georg Cantor, sets were consistent Actual Infinite collections. But he thought that all Potential Infinite collections presuppose Actual Infinite collections, much as mathematical variables range over fixed domains. So he thought of collections like that of all the sets (Cantor’s Paradox) or all the cardinal numbers as Actual Infinite but inconsistent. For more details, see Michael Hallett, Cantorian set theory and limitation of size (Clarendon Press, 1984), pp. 24–48. Of course, taking inconsistency on the chin like that is a high a price to pay for Realism (whence the foundation of mainstream mathematics is now an axiomatic set theory). But even if Potential Infinite collections do depend upon something being Actual Infinite, that might be a power (see note iv in Cantor's Paradox) or a length (see following note) rather than a collection.
......[iii] For such continua, see my “To Continue with Continuity,” Metaphysica 6 (2005): 91–109; Philip Ehrlich, “The Absolute Arithmetic Continuum and its Peircean Counterpart,” in Matthew Moore (ed.), New Essays on Peirce’s Mathematical Philosophy (Open Court, forthcoming).

No comments: