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In this article, "Cantor's Theory" refers to the pre-formal ideas about set theory introduced by Georg Cantor in the latter part of the nineteenth century. The "anti-Cantorians" are people who claim that Cantor created a fantasy world. The "Cantorians" defend Cantor's Theory. This terminology seldom appears in the mathematical literature, but it has become almost standard in Usenet discussions of Cantor's Theory. Wikipedia does not have an article with this exact name. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ... Usenet is a distributed Internet discussion system that evolved from a general purpose UUCP network of the same name. ...

The anti-Cantorians claim that while infinite sets and power sets of those infinite sets are undeniably useful abstractions, what Cantor did was to take an argument (the diagonal argument), which is perfectly valid in concrete mathematics, and recklessly apply it to the abstractions of the infinite, ultimately producing something which may have philosophical value, but no scientific value. Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ... Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...

The Cantorians (which includes almost all modern pure mathematicians) claim that since Cantor's Theory can be formalized in an (apparently) logically consistent way (e.g. ZFC), and since the study of formalizations is certainly a part of pure mathematics, there is absolutely no room for debate about Cantor's Theory. In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

Right from the start, Cantor's Theory was controversial. Witness the following quotes from contemporaries of Cantor:

"I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there" (Kronecker) Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ...

"Later generations will regard [Cantor's] set theory as a disease from which one has recovered" (Poincare) Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ...

Before Cantor came along, mathematicians generally considered mathematics to be a science, or at least, the purpose of mathematics was to provide a conceptual framework for use in the sciences. Mathematical theorems were supposed to be true and meaningful. Mathematics was supposed to have the potential to serve as a model for phenomena in the real world. The notion of infinity was at best a useful abstraction which helped mathematicians reason about the finite world. The infinite was deemed to have at most a potential existence, rather than an actual existence. Infinite redirects here, For the album by Eminem, see Infinite (album). ...

"Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already" (Poincare) Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ...

Cantor's Theory is built on the premise that the infinite is something that has an actual existence (i.e. exists as a completed totality). When Cantor introduced his ideas, there was a heated debate about whether they should be accepted as mathematics. As Kronecker pointed out, Cantor's ideas might be philosophy or theology, but it's questionable whether they are part of mathematics. To accept Cantor's ideas as mathematics changes the very definition of mathematics; mathematics changes from being a science studying patterns and phenomena related to computation, to being the examination of formal consequences of somewhat arbitrary intuitions about the infinite, with consistency being the only validating criterion.

For reasons which are not entirely clear, Cantor's ideas ultimately were accepted, and the debate over whether they should have been accepted has fallen silent within the mathematical literature. Even the constructivists and the intuitionists, who developed their schools of mathematics as a rebellion against Cantor's infinitary ideas, seem to have given up trying to argue that all mathematicians should abandon Cantor's Theory. It would appear that Hilbert's prediction has proven accurate: In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...

"No one will drive us from the paradise which Cantor created for us" (Hilbert) David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...

However, the debate has flaired up again on the internet, in newsgroups like sci.math and sci.logic. But the new debate has a different flavor than in the past. The people are different, and there's a new ingredient in the debate; the computer revolution has given the world a new paradigm for thinking about mathematics. A newsgroup is a repository, usually within the Usenet system, for messages posted from many users at different locations. ...

Most of the anti-Cantorians today are not pure mathematicians, but rather, they're people who have spent years applying mathematics in industry (typically, computers programmers and engineers). These people look back on the mathematics they learned in school, and compare it to what they've applied, and then come to the conclusion that the ideas introduced by Cantor have no relevance to the mathematics which helps us model phenomena in the real world. And with closer inspection, they often come to the conclusion that Cantor's Theory is actually quite absurd.

The anti-Cantorians generally accept the idea that the computer may be thought of as a microscope which helps us peer into a world of computation, and then mathematics is the science which studies the phenomena observed in that world of computation. If we try to merge Cantor's Theory, in which mathematics has infinite sets at its foundation, with this computation based view of mathematics, we are forced to the conclusion that while the collection of all objects in the world of computation is a countable set, and while the collection of all abstractions derived from the world of computation is a countable set, there nevertheless "exist" uncountable sets, implying the existence of a super-infinite world filled with objects that we cannot observe individually, nor talk about individually, nor even think about individually. That's the absurdity which the anti-Cantorians reject as fantasy, or even theology (i.e. the study of a world beyond what we can observe). Cantor's Theory does not show us that there is any inconsistency, nor practical disadvantage, in limiting mathematics to the study of things we can observe. In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, an uncountable set is a set which is not countable. ...

The anti-Cantorians propose that a reality criterion should be added to mathematics: we must take steps to guarantee that formal conclusions reached in the world of abstractions can be translated back into assertions about the concrete world. Infinite sets and power sets of infinite sets exist only as useful fictions (abstractions) which help us reason about the concrete reality underlying mathematics (i.e. the world or computation). Axioms and the rules of inference for abstractions must guarantee that any statement about the infinite should have implications for approximations to the infinite. It's not clear that any of the anti-Cantorians have produced a collection of axioms and rules of inference that satisfy their own criteria, and are powerful enough to do all potentially useful mathematics, though the constructivists have made impressive progress towards that goal. In mathematics, given a set S, the power set of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...

The basic anti-Cantorian argument really hasn't changed much since Cantor's time, except for the use of the computer (in the abstract) as a conceptual aid for reasoning about the foundations of mathematics. Poincare's proposed cure for the disease of Cantor's Theory, and the basic ideas behind constructivism, capture the basic idea behind the more modern anti-Cantorian arguments: The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...

"The important thing is never to introduce entities not completely definable in a finite number of words" (Poincare) Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ...

"...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ..." (Hermann Weyl) Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ...

While there is almost no debate over the validity of Cantor's Theory in the contemporary mainstream mathematical literature, nevertheless, many mathematicians informally admit that modern set theory (i.e. the formalization of Cantor's Theory) goes far beyond what could reasonably be called reality. Consider:

"Set theory is based on polite lies, things we agree on even though we know they're not true. In some ways, the foundations of mathematics has an air of unreality." (William P. Thurston) William Paul Thurston (born October 30, 1946) is an American mathematician. ...

"[The pure mathematicians] have followed a gleam that has led them out of this world...the work of the idealist who ignores reality will not survive." (Morris Kline) Morris Kline (1 May 1908 - 10 June 1992) was a Professor of Mathematics and a writer on the history, philosophy and teaching of mathematics, and of popular mathematics. ...

Most mathematicians are focused on doing mathematics, and not on worrying about the foundations of mathematics; a change in the foundations of mathematics would have essentially no affect on the mathematics they do. The vast majority of mathematics makes little or no use of the more surreal aspects of set theory.

The pure mathematicians tend to view mathematics as an art form. They seek to create beautiful theories, which may happen to be connected to reality, but only by accident. Those who apply mathematics, tend to view mathematics as a science which explores an objective reality (the world of computation). In science, truth must have observable implications, and such a "reality check" would reveal Cantor's Theory to be a pseudoscience; many of the formal theorems in Cantor's Theory (such as the Cantor's theorem asserting the existence of uncountable sets) have no observable implications. The scientists see it as their duty to try to separate reality from fantasy. The artists see the requirement that mathematical statements must have observable implications as a restriction on their intellectual freedom. Cantor's Theory is very much compatible with the view that mathematics is an art form, but it is incompatible with the view that mathematics is a science. Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. ...

The pure mathematicians tend to view the anti-Cantorians as mathematically unsophisticated, and often dismiss them as cranks. The anti-Cantorians tend to view the pure mathematicians as sophisticates who are out of touch with reality. In the Usenet debates about Cantor's Theory, neither side budges, and there seems to be no progress towards resolving the conflict. To further compound the problem, the debate tends to attract a lot of people, on both sides of the debate, who have a very limited understanding of what the issues are, and hence there is a tremendous amount of noise in the debate.

The anti-Cantorians, or at least the people who argue against Cantor's Theory on the internet, do not all speak with one voice. Almost all of the anti-Cantorians agree that Cantor's Theory is merely a collection of intuitions about infinity which have no connection to reality, but they don't all agree that the solution is to remove intuition from mathematics and treat mathematics as a science. Some of them propose replacing Cantor's intuitions with other intuitions which seem to be equally disconnected from reality (for example, often they argue that the set of all integers must be deemed to contain infinite integers). Some argue that their own intuitions are so obviously correct that their intuitions should be regarded as fundamental truths, and from that they deduce that set theory is actually inconsistent - an argument which will never have an impact on the pure mathematicians. And to be honest, the arguments of some of the loudest people who pose as anti-Cantorians seem to be private arguments which less than two people in the whole world can understand. It really isn't very surprising that most mathematicians ignore the whole debate.

For the anti-Cantorians, the bottom line is that Cantor's Theory is a mythology (a story about a world unrelated to our experience), and there is an element of social injustice in having fanatical devotees of a mythology in positions of power, deciding who gains admittance into the mathematical fraternity, deciding what ideas should be taught to students, and deciding which ideas are worthy of pursuit. The mythology is a substantial barrier to the access of mathematical knowledge for those who apply mathematics, and it diverts resources (money and manpower) away from lines of research vital to science and technology, and it is very possible that it is actually hindering progress in the field of artificial intelligence, where the ideas from the foundations of mathematics may play an essential role. If mathematicians are to accept Gauss' dictum that in mathematics, there is no true controversy, then the source of the controversy must not be accepted as part of mathematics. The word mythology (from the Greek μυϑολογία mythología, from μυϑολογειν mythologein to relate myths, from μυϑος mythos, meaning a narrative, and λογος logos, meaning speech or argument) literally means the (oral) retelling of myths – stories that a particular culture believes to be true and that use the supernatural to interpret natural events and... Artificial intelligence (also known as machine intelligence and often abbreviated as AI) is intelligence exhibited by any manufactured (i. ... Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...

For the Cantorians, the bottom line is that they have won the battle. Virtually no one in a position of authority within the mathematics community is opposed to Cantor's Theory, and the Cantorians so far have been able to take the moral high ground with respect to those outside the mathematics community who try to expose Cantor's Theory as a fraud. The Cantorians claim that by accepting Cantor's Theory, they are open minded, tolerant, and able to see the bigger picture. They claim that those who denounce Cantor's Theory are closed minded, intolerant, unimaginative, destructive cranks and malcontents who are trying to take away the freedom of humble seekers of truth. The Cantorians find great solace in the truism that just because no practical use has ever been found for Cantor's Theory, that (in itself) doesn't mean that there will never be a practical use for it.

A solution has been proposed which would satisfy the anti-Cantorians and allow the Cantorians to remain in the universities and retain the respect they desire: mathematics could be split into two disciplines - scientific mathematics (i.e. the science of phenomena observable in the world of computation), and philosophical mathematics, wherein Cantor's Theory is one of many possible formal theories of the infinite. Even though the split seems to be taking place spontaneously, if slowly, the Cantorians refuse to endorse such a split.