Cantor's diagonal argument

Cantor's diagonal argument provides a convenient proof that the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{\mathbb{N}}} of subsets of the natural numbers (also known as its power set) is not countable. More generally, it is a recurring theme in computability theory, where perhaps its most well known application is the negative solution to the halting problem.

Informal description

The original Cantor's idea was to show that the family of 0-1 infinite sequences is not countable. This is done by contradiction. If this family is countable then its members can be enumerated or enlisted. Such a list gives a table of digits, like in the following arbitrarily chosen example:

0, 1, 0, 1, 0, ...

1, 1, 1, 1, 0, ...

1, 0, 1, 0, 0, ...

Now, we construct a sequence s = (s₁, s₂, s₃, ....), which is not on the list while still, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i\in\{0,1\}} for all i. This is done as follows. Take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_1} to be different from the first digit of the first member on the list. In our example the digit is 0 (in boldface) and so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_1} is defined to be 1. Take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_2} to be different from the second digit of the second member on the list (in our example Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_2=0} ). Generally, define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_n} as different from the n-th digit of the n-th entry on the list. In other words, the sequence s = (s₁, s₂, s₃, ....) contains "the complement in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{0,1\}} " of the diagonal of our table. It follows that that the sequence s itself is not on the list, since it is different from every member by the definition. The list was supposed to contain all the 0-1 sequences. The contradiction shows that such sequences can not be enumerated (or they are not countable).

The role of the diagonal clearly explains the name of the argument.

Formal argument

To prove that the family of all subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N}} is not countable, we associate to any set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \subset \mathbb{N}} a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi : \mathbb{N} \rightarrow \{0, 1\}} by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(n) = 1} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in S} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(n) = 0} , otherwise. Conversely, every such function defines a subset. Observe also that every such function corresponds to a 0-1 sequence and vice-versa.

If power set is countable, there is a bijective map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F : \mathbb{N} \rightarrow 2^{\mathbb{N}}} , that allows us to assign an index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = F^{-1} (S)} to every subset S. In other words, all the functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi: \mathbb{N} \rightarrow \{0, 1\}} can be enumerated as $\{\phi _{1},\phi _{2},\phi _{3},\ldots \}$. Assuming this has been done, we proceed to construct a function $\psi :\mathbb {N} \rightarrow \{0,1\}$ that is not in this list. Consequently, the corresponding set, $T$ cannot be in the range of $F$.

For each $i$, either $\phi _{i}(i)=0$ or $\phi _{i}(i)=1$, and so we define $\psi (i)=1-\phi _{i}(i)$. Clearly, $\psi (i)\in \{0,1\}$ and $\psi (i)\not =\phi _{i}(i)$.

It follows that $\psi \not =\phi _{i}$ for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} , and it must therefore correspond to a set not in the range of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} . This contradiction shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{\mathbb{N}}} cannot be countable.

Application to general sets

More generally, the argument shows that a set cannot be put into one-to-one correspondence with its power set: equivalently, the cardinality of a set if strictly less than that of its power set. The argument proceeds by showing that there cannot be a surjection from a set X to the set PX of subsets of X. Suppose if possible that there were such a surjection f. Form the set

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = \{ x \in X : x \not\in f(x) \} . \, }

Since C is a subset of X then from the assumption that f is surjective there is an element c of X such that f(c) = C. We now ask whether c is in C. We find that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \in C \Leftrightarrow c \in \{ x \in X : x \not\in f(x) \} \Leftrightarrow c \not\in f(c) \Leftrightarrow c \not\in C , \,}

which is a contradiction. Hence the supposition that f is a surjection is untenable. In particular, then, f cannot be a bijection.