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From Wikipedia:

A variety of diagonal arguments are used in mathematics.

  • Cantor's diagonal argument
  • Cantor's theorem
  • Halting problem
  • Diagonal lemma

Besides the above four examples, there is another one I found in a blog. When proving that "if a sequence of measurable mappings converges in measure, then there is a subsequence converging a.e.", the construction of the subsequence is also called diagonalization:

The method for establishing this result is fairly typical of such arguments: we rely on diagonalization along with the control that Borel-Cantelli gives us. Let $f_n$ be a sequence that converges in measure to $f$. This means that for any n we have a $f_{m_n}$ with $\mu(|f_{m_n} - f| > 1/n) < 2^{-n} $. Applying Borel-Cantelli to the sequence of sets $A_n = \{x | |f_{m_n}(x) - f(x)| > 1/n\}$ yields $\mu(\limsup_m \cup_{n=m}^\infty A_n) = 0$. But this is simply saying that the set of points on which $f_{m_n}$ doesn’t converge to $f$ has measure $0$.

As someone who has not been very much exposed to "diagonalization arguments", I wonder if the examples above have something in common, so that we may answer what "diagonalization argument" is and what kinds of problems it may help to solve?

3 Answers 3