Thursday, December 09, 2010

Cantor's Paradox

This is the thirteenth of 17 posts, which are collectively Eternity, etc.
......This section is rather mathematical, but we can—indeed, should—begin with the simplest numbers, 1, 2, 3, etc. Mainstream mathematics has axiomatic set theory for a foundation, for such reasons as Cantor’s Paradox [i], and pure mathematicians are certainly free to explore any interesting formal possibilities. But we are primarily interested in possibilities insofar as they are (or might be) grounded in the God that is.
......The natural (or counting) numbers are the products of endlessly reiterating the addition of 1, starting with 1. They are clearly instantiated, because you and I are 2 people. Arithmetic is prima facie the science of such elementary metaphysical possibilities as the possibility of two individuals. Such arithmetical Realism may be difficult to justify atheistically [ii], but we may think of the natural numbers as existing amongst God’s thoughts, arising via His epistemic omnipotence from His perfect grasp of the concept of a thing, whence our informal 1, together with a concept associated with His omnipotence, such as possibility, whence informal addition and its endless reiteration. Note that Realist arithmetic can be discovered a priori under Theism because we instantiate the concept of a thing and were created in God’s image (and we might also be divinely inspired) [iii].
......The endless reiteration of the addition of 1 means that the natural numbers are (collectively) infinite. Many mathematicians have taken them to be Potential Infinite, as Aristotle put it [iv], or as J. S. Mill put it, indefinitely extensible. The addition of 1 is a definite process, but the natural numbers would have no fixed extension if the endless reiteration of the addition of 1 led to growth that could not even in principle be completed. But most of us think of arithmetic as timeless, and the modern view of the natural numbers is that they are (collectively) Actual Infinite, existing as an immutably complete collection, N = {1, 2, 3, …}. Continued...
......[i] Within an axiomatic set theory, Cantor’s Theorem says only that there is no such set of all such sets. For Cantor’s Paradox, see note iii of Divine Attributes.
......[ii] For some well-known problems, see Stewart Shapiro, Thinking about Mathematics: The philosophy of mathematics (Oxford Univ. Press, 2000), pp. 107–289; George Lakoff & Rafael E. Núñez, Where Mathematics Comes From: How the embodied mind brings mathematics into being (Basic Books, 2000), pp. 342–3.
......[iii] For more details, see Christopher Menzel, “God and Mathematical Objects,” in Russell W. Howell & W. James Bradley (eds.), Mathematics in a Postmodern Age: A Christian perspective (Eerdmans, 2001), pp. 65–97 (especially pp. 92–6).
......[iv] To see what Aristotle meant, consider an everlasting fruit-tree. The tree’s endless production of fruit is the ever-incomplete expression of its power to fruit. The total amount of fruit produced is always finite, but always increasing; it is unlimited—is Potential Infinite—because the tree’s power to fruit remains infinite. For more details, see Copan & Craig, Creation out of Nothing, pp. 200–10; Peter Fletcher, “Infinity,” in Dale Jacquette (ed.), Philosophy of Logic (Elsevier, 2007), pp. 523–85.

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