Absolute Infinite
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The Absolute Infinite is 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.
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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. [2]
Cantor also mentioned the idea in his famous letter to Richard Dedekind 28 July 1899 *:
- 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 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 certainly if we set this 0 in the first position then Ω* is still a sequence ... of which one can readily convince oneself that every number occurring in it is the [ordinal number] of the sequence of all its preceding elements. 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 belong 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. Thus 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 in a naïve sense, set theory might be said to be based on this notion. Zermelo's fix would seem to commit us to the rather mysterious notion of a proper class: a class of objects that does not have any formal existence, as an object (set), within our theory. For example, the class of all sets would be such a proper class.
Endnotes
* Ivor Grattan-Guinness has shown that this "letter" is really an amalgam by Cantor's editor Ernst Zermelo of several letters written at different times (I. Grattan-Guinness, "The rediscovery of the Cantor-Dedekind Correspondence", Jahresbericht der deutschen Mathematik-Vereinigung 76, 104-139
See also
References
- [1] Rudy Rucker, Infinity and the Mind, Princeton University Press, 1995.
- [2] Ruckerbook Mind Tools
- [3] Heijenoort 1967
- [4] Moore, A.W. The Infinite, New York, Routledge, 1990
- [5] Moore, A.W. "Set Theory, Skolem's Paradox and the Tractatus", Analysis 1985, 45
- [6] G. Cantor, 1932. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. E. Zermelo, Ed. Berlin: Springer; reprinted Hildesheim: Olms, 1962; Berlin/Heidelberg/New York: Springer, 1980.nl:Absoluut oneindige