Cantor's Diagonal Argument - Cantor's Paradise - Medium

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In addition to his inventions of set theory and transfinite numbers, Georg Cantor (1845–1918) is remembered as the brilliant inventor of the popular diagonalization argument later employed by both…
Proof Theory

"Diagonalization seems to show that there is an inexhaustibility phenomenon for definability similar to that for provability" — Franzén (2004)

In addition to his inventions of set theory and transfinite numbers, Georg Cantor (1845–1918) is remembered as the brilliant inventor of the popular diagonalization argument later employed by both Kurt Gödel (1906–1978) and Alan Turing (1912–1954) in their most famous papers.

The Diagonal Argument

In set theory, the diagonal argument is a mathematical argument originally employed by Cantor to show that

"There are infinite sets which cannot be put into one-to-one correspondence with the infinite set of the natural numbers" — Georg Cantor, 1891

Mathematically, Torkel Franzén (1950–2006) describes the diagonal construction as follows in his wonderful treatise on Gödel's incompleteness theorems, entitled Inexhaustability* (2005):

Given a relation R(a, b) between elements a and b in some set A, if we define the property P of elements in A to hold if and only if R(a, a) does not hold, this property P is not identical with the property Rₐ for any a, where Rₐ(b) is defined to hold if and only if R(a, b) holds.

An alternative form, also provided by Torkel Franzén in his more accessible book Gödel's Theorem* (2004) is:

By enumerating the conditions F₁(x), F₂(x), ... which define sets in some particular language L, we can define a new set D by diagonalization, as the set of k such that Fk(k) is not true. If Mk is the set {x|Fk(x)} we see that D is different from all the Mk, since for every k, k ∈ Mk if and only if k ∉ D.

To demonstrate its function and utility, consider the following three examples, by Cantor (1891), Gödel (1931) and Turing (1937).

The Uncountability of Real Numbers (Cantor, 1891)

Cantor first demonstrated the uncountability of real numbers using a topological proof, in the 1874 paper

Cantor, G. (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen". Journal für…
Jørgen Veisdal
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