Absolute infinite

The absolute infinite (symbol: Ω), in context often called "absolute", is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite. Cantor linked the absolute infinite with God,^{[1][2]: 175 [3]: 556} and believed that it had various mathematical properties, including the reflection principle: every property of the absolute infinite is also held by some smaller object.^{[4][clarification needed]}

Cantor's view

Cantor said:

The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or order type. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.^[5]

Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original):^[7]

A multiplicity [he appears to mean what we now call a set] 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

Main article: Burali-Forti paradox

The idea that the collection of all ordinal numbers cannot logically exist seems paradoxical to many. This is related to the Burali-Forti's paradox which implies 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.

While this solves the logical problem, one could argue that 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 such as New Foundations by Willard Van Orman Quine.

See also

• Actual infinity
• Limitation of size
• Monadology
• Reflection principle
• Absolute (philosophy)
• Ineffability

Notes

1. §3.2, Ignacio Jané (May 1995). "The role of the absolute infinite in Cantor's conception of set". Erkenntnis. 42 (3): 375–402. doi:10.1007/BF01129011. JSTOR 20012628. S2CID 122487235. Cantor (1) took the absolute to be a manifestation of God [...] When the absolute is first introduced in Grundlagen, it is linked to God: "the true infinite or absolute, which is in God, admits no kind of determination" (Cantor 1883b, p. 175) This is not an incidental remark, for Cantor is very explicit and insistent about the relation between the absolute and God.
2. Georg Cantor (1932). Ernst Zermelo (ed.). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Berlin: Verlag von Julius Springer. Cited as Cantor 1883b by Jané; with biography by Adolf Fraenkel; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, ISBN 3-540-09849-6.
3. Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. Original article.
4. Infinity: New Research and Frontiers by Michael Heller and W. Hugh Woodin (2011), p. 11.
5. https://www.uni-siegen.de/fb6/phima/lehre/phima10/quellentexte/handout-phima-teil4b.pdf
   Translated quote from German:

   Es wurde das Aktual-Unendliche (A-U.) nach drei Beziehungen unterschieden: erstens, sofern es in der höchsten Vollkommenheit, im völlig unabhängigen außerweltlichen Sein, in Deo realisiert ist, wo ich es Absolut Unendliches oder kurzweg Absolutes nenne; zweitens, sofern es in der abhängigen, kreatürlichen Welt vertreten ist; drittens, sofern es als mathematische Größe, Zahl oder Ordnungstypus vom Denken in abstracto aufgefaßt werden kann. In den beiden letzten Beziehungen, wo es offenbar als beschränktes, noch weiterer Vermehrung fähiges und insofern dem Endlichen verwandtes A.-U. sich darstellt, nenne ich es Transfinitum und setze es dem Absoluten strengstens entgegen.

   [Ca-a,^[2] p. 378].
6. 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.
7. Gesammelte Abhandlungen,^[2] 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-Guinness has discovered,^[6] 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.

Bibliography

• The role of the absolute infinite in Cantor's conception of set
• Infinity and the Mind, Rudy Rucker, Princeton, New Jersey: Princeton University Press, 1995, ISBN 0-691-00172-3; orig. pub. Boston: Birkhäuser, 1982, ISBN 3-7643-3034-1.
• The Infinite, A. W. Moore, London, New York: Routledge, 1990, ISBN 0-415-03307-1.
• Set Theory, Skolem's Paradox and the Tractatus, A. W. Moore, Analysis 45, #1 (January 1985), pp. 13–20.