Philosophical objections to 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 first proof 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 (or greater cardinality) than the set of natural numbers N = {1, 2, 3, …}. This larger set consists of the elements (x1, x2, x3, …), where each xn is either m or w. Each of these elements corresponds to a subset of N—namely, the element (x1, x2, x3, …) corresponds to {n ∈ N: xn = w}. 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(N), the power set of N.
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