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

Table 1.2from potential infinity to actual infinity 41Levels of Infinity Epistemology OntologyAbsolute Infinity (Actual Infinity) Intuition () BeingGeorg Cantor: tNicholas of Cusa: “coincidentiaoppositorum”The Areopagite, Gregory of Nyssa:apophatic theologyThe Transfinite (Actual Infinity: a 0 ,..., Discursive quantitative Becominga 1 ,...,a w ): Georg Cantorrationality ()Potential Infinity: AristotleDiscursive quantitative BecomingFiniteness (), to be avoided due tolack of form: rationality ()Sensual experience(, )Phenomenal worldof senses1.4.2.4.3 Resolving the Antinomy. Georg Cantor resolved this contradiction by claimingthat t, the absolute infinity, cannot be an object of quantitative, discursive, rationaloperation. It cannot be understood by logical discernment, but only by intuitiveinsight. 107 Furthermore, it cannot be recognized, for it can only be accepted withoutany further discursive rational activity and logical discernment. 108 “Logical discernment”in this context intended to show that he did not conceive of t = {a 0, ...,a 1 ,...,a n }as a set. Instead, he called it an “inconsistent plurality,” to which his theorem ofthe cardinality of sets could not be applied without creating logical inconsistencies. 109Hence, he avoided the logical contradictions of T by excluding it from being part ofsets. Instead, he created another type of sets, which he called the “inconsistent plurality.”However, this logical differentiation between sets and inconsistent pluralities wasmore than just a formal logical operation.This t, Georg Cantor claimed, is God, the creative source of all quantities existingin the world, and an intuitive insight of God is possible. It was the transformativeexperience of this t that helped Cantor, according to his own words, to find thetransfinite numbers with all their strange mathematical properties. 110The following Table 1.2 summarizes all the epistemological, ontological, and mathematicallevels of infinity and the way they are related to each other:107 Georg Cantor made an allusion to this kind of intuitive insight in a letter to Philip Jourdain from 1903: “I have20 years ago intuitively realized (when I discovered the Alefs themselves) the undoubtedly correct theorem,that except the Alefs there are no transfinite cardinal numbers” (Bandmann 1992, p. 282).108 “The absolute infinity can only be accepted, but not be recognized, not even nearly recognized” (Bandmann1992, p. 285).109 More explicitly, Georg Cantor worked with two theorems to avoid the contradiction. Theorem A: The systemT of all Å’s is in its extension similar to and is for this reason also an inconsistent plurality. Theorem B:The system T of all alephs is noting else as the system of all transfinite cardinal numbers (Bandmann 1992,p. 281). In a letter from 1897 to David Hilbert, Georg Cantor introduced this distinction between normal setsand inconsistent pluralities. He wrote, “The totality of all Alefs is a totality which cannot be conceived as adistinct well-defined set. If this was the case, it would entail another distinct Alef following this totality, whichwould at the same time belong to this totality and not belong to it. This would be a contradiction. Totalities,which cannot be conceived from our perspective as sets ( . . . ) I called already years ago absolute infinitetotalities and have distinguished them very clearly from the transfinite sets” (Bandmann 1992, p. 287).110 For the purposes of this chapter, the important publications, in which Cantor develops his thoughts aboutinfinity, are Cantor (1883, 1932).