- Absolute Infinite
The Absolute Infinite is
mathematician Georg Cantor's concept of an " infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God. [§3.2, cite journal
title=The role of the absolute infinite in Cantor's conception of set
volume=42 | issue=3 | month=May | year=1995 | pages=375–402 | doi=10.1007/BF01129011] He held that the Absolute
Infinitehad various mathematicalproperties, including that every property of the Absolute Infinite is also held by some smaller objectFact|date=November 2007.
Cantor is quoted as saying:
The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, "in Deo", where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it "in abstracto" as a mathematical magnitude, number or order type. [Quoted in "Mind Tools: The Five Levels of Mathematical Reality", Rudy Rucker, Boston: Houghton Mifflin, 1987; ISBN 0395383153.]
Cantor also mentioned the idea in his letters to
Richard Dedekind(text in square brackets not present in original): ["Gesammelte Abhandlungen"4, Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962, pp. 443–447; translated into English in "From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931", ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as Ivor Grattan-Guinnesshas discovered5, this is in fact an amalgamation by Cantor's editor, Ernst Zermelo, of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August 3.]
Now I envisage the system of all [ordinal] numbers and denote it "Ω".
The system "Ω" in its natural ordering according to magnitude is a "sequence".
Now let us adjoin 0 as an additional element to this sequence, and place it, obviously, in the first position; then we obtain a sequence "Ω′"::0, 1, 2, 3, … ω0, ω0+1, …, γ, …
of which one can readily convince oneself that every number γ occurring in it is the type [i.e., order-type] of the sequence of all its preceding elements (including 0). (The sequence "Ω" has this property first for ω0+1. [ω0+1 should be ω0.] )
Now "Ω′" (and therefore also "Ω") cannot be a consistent multiplicity. For if "Ω′" were consistent, then as a well-ordered set, a number "δ" would correspond to it which would be greater than all numbers of the system "Ω"; the number "δ", however, also belongs to the system "Ω", because it comprises all numbers. Thus "δ" would be greater than "δ", which is a contradiction. Therefore:
"The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity."
The Burali-Forti paradox
The idea that the collection of all ordinal numbers cannot logically exist seems
paradoxicalto many. This is related to Cesare Burali-Forti's "paradox" that there can be no greatest ordinal number. All of these problems can be traced back to the idea that, for every property that can be logically defined, there exists a set of all objects that have that property. However, as in Cantor's argument (above), this idea leads to difficulties.
More generally, as noted by
A.W. Moore, there can be no end to the process of set formation, and thus no such thing as the "totality of all sets", or the "set hierarchy". Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.
A standard solution to this problem is found in Zermelo's set theory, which does not allow the unrestricted formation of sets from arbitrary properties. Rather, we may form the set of all objects that have a given property "and lie in some given set" (Zermelo's Axiom of Separation). This allows for the formation of sets based on properties, in a limited sense, while (hopefully) preserving the consistency of the theory.
However, while this neatly solves the logical problem, the philosophical problem remains. It seems natural that a set of individuals ought to exist, so long as the individuals exist. Indeed,
naive set theorymight be said to be based on this notion. Although Zermelo's fix allows a class to describe arbitrary (possibly "large") entities, these predicates of the meta-languagemay have no formal existence (i.e., as a set) within the theory. For example, the class of all sets would be a proper class. This is philosophically unsatisfying to some and has motivated additional work in set theoryand other methods of formalizing the foundations of mathematics.
Class (set theory)
Limitation of size
References and further reading
- ^"Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Georg Cantor, ed. Ernst Zermelo, with biography by Adolf Fraenkel; orig. pub. Berlin: Verlag von Julius Springer, 1932; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, ISBN 3540098496.
- ^ [http://www.digizeitschriften.de/home/services/pdfterms/?ID=516899 The Rediscovery of the Cantor-Dedekind Correspondence] , I. Grattan-Guinness, "Jahresbericht der Deutschen Mathematiker-Vereinigung" 76 (1974/75), pp. 104–139, at p. 126 ff.
- "Infinity and the Mind",
Rudy Rucker, Princeton, New Jersey: Princeton University Press, 1995, ISBN 0691001723; orig. pub. Boston: Birkhäuser, 1982, ISBN 3764330341.
- "The Infinite", A. W. Moore, London, New York: Routledge, 1990, ISBN 0415033071.
- [http://links.jstor.org/sici?sici=0003-2638%28198501%2945%3A1%3C13%3ASTSPAT%3E2.0.CO%3B2-%23 Set Theory, Skolem's Paradox and the "Tractatus"] , A. W. Moore, "Analysis" 45, #1 (January 1985), pp. 13–20.
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