Controversy over Cantor's theory
Encyclopedia
In mathematical logic
, the theory of infinite sets was first developed by Georg Cantor
. Although this work has found wide acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers.
Cantor's theorem
that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}, has probably attracted more hostility than any other mathematical argument, before or since, with the exception of Hilbert's introduction of completely non-constructive proofs of existence some decades later. (Cantor's work gave rise to some flashy soundbites from Kronecker et al.; Hilbert's sparked a blazing row which gave rise to constructivism in mathematics and mathematical logic.) Logician has commented on the energy devoted to refuting this "harmless little argument", asking, "what had it done to anyone to make them angry with it?"
s), which has a larger number of elements, or as he puts it, has a greater 'power' (Mächtigkeit), than the infinite set of finite whole numbers
{1, 2, 3, ...}.
There are a number of steps implicit in his argument, as follows:
claimed: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." Many mathematicians agreed with Kronecker
that the completed infinite may be part of philosophy
or theology
, but that it has no proper place in mathematics.
Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence. "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'. In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets.
Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert
, amongst others. Hilbert predicted: "No one will drive us from the paradise which Cantor created for us". To which Wittgenstein
replied "if one person can see it as a paradise of mathematicians, why should not another see it as a joke?". Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism
and intuitionism
.
. It is a generally recognized view by logicians that this axiom is not a logical truth
. Indeed, as Mark Sainsbury
has argued "there is room for doubt about whether it is a contingent truth, since it is an open question whether the universe is finite or infinite". Bertrand Russell
for many years tried to establish a foundation for mathematics that did not rely on this axiom. Mayberry has noted that "The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them – indeed, the most important of them, namely Cantor's axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all".
Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl
wrote:
Richard Arthur, philosopher and expert on Leibniz, has argued that Cantor's appeal to the idea of an actual infinite (formally captured by the axiom of infinity) is philosophically unjustified. Arthur argues that Leibniz' idea of a "syncategorematic
" but actual infinity is philosophically more appealing. (See external link below for one of his papers).
The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics (for example, that includes real analysis
).
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, the theory of infinite sets was first developed by 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,...
. Although this work has found wide acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers.
Cantor's theorem
Cantor's theorem
In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}, has probably attracted more hostility than any other mathematical argument, before or since, with the exception of Hilbert's introduction of completely non-constructive proofs of existence some decades later. (Cantor's work gave rise to some flashy soundbites from Kronecker et al.; Hilbert's sparked a blazing row which gave rise to constructivism in mathematics and mathematical logic.) Logician has commented on the energy devoted to refuting this "harmless little argument", asking, "what had it done to anyone to make them angry with it?"
Cantor's argument
Cantor's 1891 argument is that there exists an infinite set (which he identifies with the set of real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s), which has a larger number of elements, or as he puts it, has a greater 'power' (Mächtigkeit), than the infinite set of finite whole numbers
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
{1, 2, 3, ...}.
There are a number of steps implicit in his argument, as follows:
- That the elements of no set can be put into one-to-one correspondence with all of its subsets. This is known as Cantor's theoremCantor's theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
. It depends on very few of the assumptions of set theorySet theorySet 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, as John P. Mayberry puts it, is a "simple and beautiful argument" that is "pregnant with consequences". Few have seriously questioned this step of the argument.
- That the concept of "having the same number" can be captured by the idea of one-to-one correspondence. This (purely definitional) assumption is sometimes known as Hume's principleHume's principleHume's Principle or HP—the terms were coined by George Boolos—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence between the Fs and the Gs. HP can be stated formally in systems of second-order logic...
. As FregeGottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
says, "If a waiter wishes to be certain of laying exactly as many knives on a table as plates, he has no need to count either of them; all he has to do is to lay immediately to the right of every plate a knife, taking care that every knife on the table lies immediately to the right of a plate. Plates and knives are thus correlated one to one" (1884, tr. 1953, §70). Sets in such a correlation are often called equipollent, and the correlation itself is called a bijective function.
- That there exists at least one infinite set of things, usually identified with the set of all finite whole numbers or "natural numbers". This assumption (not formally specified by Cantor) is captured in formal set theory by the axiom of infinityAxiom of infinityIn axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
. This assumption allows us to prove, together with Cantor's theorem, that there exists at least one set that cannot be correlated one-to-one with all its subsets. It does not prove, however, that there in fact exists any set corresponding to "all the subsets".
- That there does indeed exist a set of all subsets of the natural numbers is captured in formal set theory by the power set axiomAxiom of power setIn mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:...
, which says that for every set there is a set of all of its subsets. (For example, the subsets of the set {a, b} are { }, {a}, {b}, and {a, b}). This allows us to prove that there exists an infinite set which is not equipollent with the set of natural numbers. The set N of natural numbers exists (by the axiom of infinity), and so does the set R of all its subsets (by the power set axiom). By Cantor's theorem, R cannot be one-to-one correlated with N, and by Cantor's definition of number or "power", it follows that R has a different number than N. It does not prove, however, that the number of elements in R is in fact greater than the number of elements in N, for only the notion of two sets having different power has been specified; given two sets of different power, nothing so far has specified which of the two is greater.
- Cantor presented a well-ordered sequence of cardinal numbers, the alephs, attempted to prove that the power of every well-defined set ("consistent multiplicity") is an aleph; therefore the ordering relation among alephs determines an order among the size of sets. However this proof was flawed, and as Zermelo wrote, "It is precisely at this point that the weakness of the proof sketched here lies… It is precisely doubts of this kind that impelled ... [my own] proof of the well-ordering theorem purely upon the axiom of choice…"
- The assumption of the axiom of choice was later shown unnecessary by the Cantor-Bernstein-Schröder theorem, which makes use of the notion of injective functionInjective functionIn mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
s from one set to another—a correlation which associates different elements of the former set with different elements of the latter set. The theorem shows that if there is an injective functionInjective functionIn mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
from set A to set B, and another one from B to A, then there is a bijective function from A to B, and so the sets are equipollent, by the definition we have adopted. Thus it makes sense to say that the power of one set is at least as large as another if there is an injection from the latter to the former, and this will be consistent with our definition of having the same power. Since the set of natural numbers can be embedded in its power set, but the two sets are not of the same power, as shown, we can therefore say the set of natural numbers is of lesser power than its power set. However, despite its avoidance of the axiom of choice, the proof of the Cantor-Bernstein-Schröder theorem is still not constructiveConstructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object...
, in that it does not produce a concrete bijection in general.
Reception of the argument
At the start, Cantor's Theory was controversial among mathematicians and (later) philosophers. As Leopold KroneckerLeopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
claimed: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." Many mathematicians agreed with Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
that the completed infinite may be part of philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
or theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...
, but that it has no proper place in mathematics.
Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence. "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'. In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets.
Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, amongst others. Hilbert predicted: "No one will drive us from the paradise which Cantor created for us". To which Wittgenstein
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
replied "if one person can see it as a paradise of mathematicians, why should not another see it as a joke?". Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
and intuitionism
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
.
Objection to the axiom of infinity
A common objection to Cantor's theory of infinite number involves the axiom of infinityAxiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
. It is a generally recognized view by logicians that this axiom is not a logical truth
Logical truth
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.Logical...
. Indeed, as Mark Sainsbury
Mark Sainsbury
Mark Sainsbury may refer to:*Mark Sainsbury , United Kingdom philosopher*Mark Sainsbury , New Zealand current affairs presenter...
has argued "there is room for doubt about whether it is a contingent truth, since it is an open question whether the universe is finite or infinite". Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
for many years tried to establish a foundation for mathematics that did not rely on this axiom. Mayberry has noted that "The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them – indeed, the most important of them, namely Cantor's axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all".
Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
wrote:
Richard Arthur, philosopher and expert on Leibniz, has argued that Cantor's appeal to the idea of an actual infinite (formally captured by the axiom of infinity) is philosophically unjustified. Arthur argues that Leibniz' idea of a "syncategorematic
Syncategorematic term
In scholastic logic, a syncategorematic term is a word that cannot serve as the subject or the predicate of a proposition, and thus cannot stand for any of Aristotle's categories, but can be used with other terms to form a proposition...
" but actual infinity is philosophically more appealing. (See external link below for one of his papers).
The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics (for example, that includes real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
).