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The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God.^[1] 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^{[citation needed]}. Image File history File links Broom_icon. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... For other uses, see Infinity (disambiguation). ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... This article discusses the term God in the context of monotheism and henotheism. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ...

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 letters to Richard Dedekind (text in square brackets not present in original):^[3] Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ...

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". Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...

…

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 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. In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naïvely constructing the set of all ordinal numbers leads to a contradiction and therefore shows an antinomy in a system that allows its construction. ... For other meanings of Paradox, see Paradox (disambiguation). ... In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naïvely constructing the set of all ordinal numbers leads to a contradiction and therefore shows an antinomy in a system that allows its construction. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...

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. Arthur William Moore, Manx historian, folklorist and politician Adrian William Moore, philosopher at University of Oxford Category: ... This article is about sets in mathematics. ... In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ...

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. Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set 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. This article is about the mathematical topic. ... Look up class in Wiktionary, the free dictionary. ... Metalanguage can refer to: An intermediate step in the compilation/assembly/interpreting process. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

See also

• Reflection principle
• Class (set theory)
• Infinity
• Limitation of size
• The Ultimate

In set theory, a branch of mathematics, a reflection principle says that we can find sets that resemble the class of all sets. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... For other uses, see Infinity (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... The Ultimate is a general term embracing the concept of an ultimate supernatural reality which transcends material reality and from which, according to a broad spectrum of Eastern philosophies and religions, material reality derives. ...

References and further reading

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. 
2. ^ Quoted in Mind Tools: The Five Levels of Mathematical Reality, Rudy Rucker, Boston: Houghton Mifflin, 1987; ISBN 0395383153.
3. ^ 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-Guinness has discovered^[5], 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.

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