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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. He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller object.

Cantor's view

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.

Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original):

A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a "sequence".

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 paradoxical to 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 theory might 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-language may 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 theory and other methods of formalizing the foundations of mathematics.

See also

References and further reading

  1. ^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.
  2. ^ 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.
  3. Infinity and the Mind, Rudy Rucker, Princeton, New Jersey: Princeton University Press, 1995, ISBN 0691001723; orig. pub. Boston: Birkhäuser, 1982, ISBN 3764330341.
  4. The Infinite, A. W. Moore, London, New York: Routledge, 1990, ISBN 0415033071.
  5. Set Theory, Skolem's Paradox and the Tractatus, A. W. Moore, Analysis 45, #1 (January 1985), pp. 13–20.

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