Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

eferences 47between theology on the one hand and mathematics on the other. As a result of a longhistorical process, we can identify certain current traits of the concept of infinity inmathematics and theology:1. Even though the infinity of God was introduced into Christian theology in order toovercome the limitations of ancient Greek metaphysics, and even though the progress inmathematics was stimulated by the theological idea of infinity, identifying infinity as animportant aspect of God is nearly abandoned in contemporary theology. If it still exists,it is only used more or less in a metaphorical way.2. Since the Enlightenment, however, starting with Schleiermacher, infinity in the realm ofreligion and theology has been serving a different function. As we have seen, the idea ofinfinity has played the role of discovering, constituting, and transcending the religiousSelf. The concept of infinity has migrated from a property of God to a factor in religiousanthropology.3. This loss of the concept of infinity in theology as a trait of God is unacceptable, althoughthere are good reasons not simply to use Cantor’s ideas for a theological concept ofGod’s infinity, because that entails thinking about it in terms of quantification. On theother hand, a purely qualitative, metaphorical, or symbolic way of talking about God’sinfinity, as is the case in many contemporary theologies, is also unsatisfactory. For myunderstanding, the theological task today is not so much to understand infinity as aproperty of God, as well as how to describe this property, but rather to understand Godas the creator in his infinite creativity. Any robust theology of creation has to come upwith an exploration of God’s infinite creative possibilities.4. In the realm of mathematics, Cantor’s work on infinity in set theory is still of paramountimportance, with all its implications on logic and the continuum. However, it would befar more interesting and forward-looking to follow the line of thought introduced bythe Hilbert program. It is commonly argued that Hilbert’s program failed as a result ofGödel’s arguments. He argued against the simultaneous validity of completeness andconsistency of arithmetic structures with a certain degree of complexity. Nevertheless,only a few years later Gerhard Gentzen, a scholar of the Hilbert school, has shown –going beyond Gödel – that with stronger proof methods, such as “transfinite induction,”the Hilbert program can be carried out further, going beyond the limitations for whichGödel had argued (Gentzen 1936a, 1936b). It seems to me that following the way thatGentzen has proposed could lead to new insights in infinite levels of complexity inarithmetic – analogs to Cantor’s levels of infinity in set theory.ReferencesAquinas, T. 1934. Summa Theologica, bd. 1, Gottes Dasein und Wesen. Salzburg, Leipzig, andRegensburg: Verlag Anton Pustet.Aristotle. 1993. Physics, books I–IV, Loeb Classical Library, G. P. Goold (ed.). Translated by P. H.Wicksteed and F. M. Cornford. Cambridge: Harvard University Press.Aristotle. 1997. Metaphysics, books X–XIV, Loeb Classical Library, G. P. Goold (ed.). Translated byC. G. Armstrong. Cambridge: Harvard University Press.