Controversy over Cantor's theory
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers.
Cantor's theorem implies that there are sets having cardinality greater than the infinite cardinality of the set of natural numbers. Cantor's argument for this theorem is presented with one small change. This argument can be improved by using a definition he gave later. The resulting argument uses only five axioms of set theory.
Cantor's set theory was controversial at the start, but later became largely accepted. In particular, there have been objections to its use of infinite sets.
Cantor's argument
that infinite sets can have different cardinalities was published in 1874. This proof demonstrates that the set of natural numbers and the set of real numbers have different cardinalities. It uses the theorem that a bounded increasing sequence of real numbers has a limit, which can be proved by using Cantor's or Richard Dedekind's construction of the irrational numbers. Because Leopold Kronecker did not accept these constructions, Cantor was motivated to develop a new proof.In 1891, he published "a much simpler proof... which does not depend on considering the irrational numbers." His new proof uses his diagonal argument to prove that there exists an infinite set with a larger number of elements than the set of natural numbers N = . This larger set consists of the elements, where each xn is either m or w. Each of these elements corresponds to a subset of N—namely, the element corresponds to. So Cantor's argument implies that the set of all subsets of N has greater cardinality than N. The set of all subsets of N is denoted by P, the power set of N.
Cantor generalized his argument to an arbitrary set A and the set consisting of all functions from A to. Each of these functions corresponds to a subset of A, so his generalized argument implies the theorem: The power set P has greater cardinality than A. This is known as Cantor's theorem.
The argument below is a modern version of Cantor's argument that uses power sets. By presenting a modern argument, it is possible to see which assumptions of axiomatic set theory are used. The first part of the argument proves that N and P have different cardinalities:
- There exists at least one infinite set. This assumption is captured in formal set theory by the axiom of infinity. This axiom implies that N, the set of all natural numbers, exists.
- P, the set of all subsets of N, exists. In formal set theory, this is implied by the power set axiom, which says that for every set there is a set of all of its subsets.
- The concept of "having the same number" or "having the same cardinality" can be captured by the idea of one-to-one correspondence. This assumption is sometimes known as Hume's principle. As Frege said, "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." Sets in such a correlation are called equinumerous, and the correlation is called a one-to-one correspondence.
- A set cannot be put into one-to-one correspondence with its power set. This implies that N and P have different cardinalities. It depends on very few assumptions of set theory, and, as John P. Mayberry puts it, is a "simple and beautiful argument" that is "pregnant with consequences". Here is the argument:
Around 1895, Cantor began to regard the well-ordering principle as a theorem and attempted to prove it. In 1895, Cantor also gave a new definition of "greater than" that correctly defines this concept without the aid of his well-ordering principle. By using Cantor's new definition, the modern argument that P has greater cardinality than N can be completed using weaker assumptions than his original argument:
- The concept of "having greater cardinality" can be captured by Cantor's 1895 definition: B has greater cardinality than A if A is equinumerous with a subset of B, and B is not equinumerous with a subset of A. Clause says B is at least as large as A, which is consistent with our definition of "having the same cardinality". Clause implies that the case where A and B are equinumerous with a subset of the other set is false. Since clause says that A is not at least as large as B, the two clauses together say that B is larger than A.
- The power set has greater cardinality than which implies that P has greater cardinality than N. Here is the proof:
- Define the subset Define which maps onto Since implies is a one-to-one correspondence from to Therefore, is equinumerous with a subset of
- Using proof by contradiction, assume that a subset of is equinumerous with Then there is a one-to-one correspondence from to Define from to if then if then Since maps onto maps onto contradicting the theorem above stating that a function from to is not onto. Therefore, is not equinumerous with a subset of
Reception of the argument
Initially, Cantor's theory was controversial among mathematicians and philosophers. As Leopold Kronecker 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. 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?" Others have also taken issue with Cantor's proof regarding the cardinality of the power set. Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday 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". Carl Friedrich 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?" The rejection of Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism and intuitionism.
Objection to the axiom of infinity
A common objection to Cantor's theory of infinite number involves the axiom of infinity. 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:
The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics.