The Article
Cantor's article begins with a discussion of the real algebraic numbers, and a statement of his first theorem: The collection of real algebraic numbers can be put into one-to-one correspondence with the collection of positive integers. Cantor restates this theorem in terms more familiar to mathematicians of his time: The collection of real algebraic numbers can be written as an infinite sequence in which each number appears only once.
Next Cantor states his second theorem: Given any sequence of real numbers x1, x2, x3, … and any interval, one can determine numbers in that are not contained in the given sequence.
Cantor observes that combining his two theorems yields a new proof of the theorem: Every interval contains infinitely many transcendental numbers. This theorem was first proved by Joseph Liouville.
He then remarks that his second theorem is:
- the reason why collections of real numbers forming a so-called continuum (such as, all real numbers which are ≥ 0 and ≤ 1) cannot correspond one-to-one with the collection (ν) ; thus I have found the clear difference between a so-called continuum and a collection like the totality of real algebraic numbers.
The first half of this remark is Cantor's uncountability theorem. Cantor does not explicitly prove this theorem, probably because it follows easily from his second theorem. To prove it, use proof by contradiction. Assume that the interval can be put into one-to-one correspondence with the set of positive integers, or equivalently: The real numbers in can be written as a sequence in which each real number appears only once. Applying Cantor's second theorem to this sequence and produces a real number in that does not belong to the sequence. This contradicts our original assumption, and proves the uncountability theorem.
Note how Cantor's second theorem separates the constructive content of his work from the proof by contradiction needed to establish uncountability.
Read more about this topic: Cantor's First Uncountability Proof
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