The Absolute Infinite (''symbol'': Ω) is an extension of the idea of infinity proposed by

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...

. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or

transfinite Transfinite may refer to: * Transfinite number, a number larger than all finite numbers, yet not absolutely infinite * Transfinite induction, an extension of mathematical induction to well-ordered sets ** Transfinite recursion Transfinite inducti ...

. Cantor linked the Absolute Infinite with

God In monotheistic thought, God is usually viewed as the supreme being, creator, and principal object of faith. Swinburne, R.G. "God" in Honderich, Ted. (ed)''The Oxford Companion to Philosophy'', Oxford University Press, 1995. God is typically ...

, Cited as ''Cantor 1883b'' by Jané; with biography by Adolf Fraenkel; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, . Original article. and believed that it had various

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.''Infinity: New Research and Frontiers'' by Michael Heller and W. Hugh Woodin (2011)
p. 11

Cantor's view

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

The Burali-Forti paradox

The idea that the collection of all ordinal numbers cannot logically exist seems

paradoxical A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

to many. This is related to Cesare Burali-Forti's "paradox" which states 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 In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...

). 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 In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quot ...

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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

and other methods of formalizing the foundations of mathematics such as New Foundations by

Willard Van Orman Quine Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...

Notes

Bibliography

''The role of the absolute infinite in Cantor's conception of set''
* ''Infinity and the Mind'',

Rudy Rucker Rudolf von Bitter Rucker (; born March 22, 1946) is an American mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known f ...

, Princeton, New Jersey: Princeton University Press, 1995, ; orig. pub. Boston: Birkhäuser, 1982, . * ''The Infinite'', A. W. Moore, London, New York: Routledge, 1990, .
Set Theory, Skolem's Paradox and the ''Tractatus''
A. W. Moore, ''Analysis'' 45, #1 (January 1985), pp. 13–20. {{Infinity Philosophy of mathematics Infinity Superlatives in religion Conceptions of God

Cantor's view

The Burali-Forti paradox

See also

Notes

Bibliography