Absolute Infinite
Encyclopedia
The Absolute Infinite is mathematician
Georg Cantor
's concept of an "infinity
" that transcended the transfinite number
s. 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.
Cantor also mentioned the idea in his letters to Richard Dedekind
(text in square brackets not present in original):
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
.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
's concept of an "infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
" that transcended the transfinite number
Transfinite number
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these...
s. Cantor equated the Absolute Infinite with God
God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....
. 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.
Cantor's view
Cantor is quoted as saying:Cantor also mentioned the idea in his letters to Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
(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 to many. This is related to Cesare Burali-Forti's "paradox"Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
that there can be no greatest ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
. 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
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
, 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
Axiom schema of specification
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory...
). 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
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
might be said to be based on this notion. Although Zermelo's fix allows a class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
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
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and other methods of formalizing the foundations of mathematics such as New Foundations by Willard Van Orman Quine
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
.
See also
- Actual infinityActual infinityActual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural...
- Class (set theory)Class (set theory)In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
- InfinityInfinityInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
- Limitation of sizeLimitation of sizeIn the philosophy of mathematics, specifically the philosophical foundations of set theory, limitation of size is a concept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor's paradox. It identifies certain "inconsistent multiplicities", in Cantor's terminology, that cannot be sets...
- MonadologyMonadologyThe Monadology is one of Gottfried Leibniz’s best known works representing his later philosophy. It is a short text which sketches in some 90 paragraphs a metaphysics of simple substances, or monads.- Text :...
- Reflection principleReflection principleIn set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble"...
- The Ultimate (philosophy)