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 D [delta] would belong to it which would be greater than all numbers of the system Ω; the number D, however, also belongs to the system Ω, because it comprises all numbers. Thus D would be greater than D, which is a contradiction. Thus the system Ω of all ordinal numbers is an inconsistent, absolutely infinite multiplicity."
The Burali-Forti paradox
This seems paradoxical, and is closely related to Cesare Burali-Forti's "paradox" that there can be no greatest ordinal number. There is a quick fix in Zermelo's system by his Axiom of Separation, which stipulates that sets cannot be independently defined by any arbitrary logically definable notion, but must be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers".
But it is a philosophical problem. It is a problem for the view that a set of individuals must exist, so long as the individuals exist. Moreover, Zermelo's fix commits us to rather mysterious objects called "proper classes". The expression "x is a set" is the name of such a class, what sort of object is it? So is the object named by "x is a thing" a thing or not?
As A.W. Moore notes, 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.
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
- The Absolute
- class (set theory)
- Gödel's ontological proof
- infinity
- reflection principle
- The Ultimate
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