As Cantor diagonalization are two of Georg Cantor called developed Diagonalisierungsbeweisverfahren:
- Cantor's first diagonal argument is a mathematical proof procedure with which one can show that the set of rational numbers is countable .
- Cantor's second diagonal argument is a mathematical proof that the set of real numbers (also called the continuum ) is uncountable . This proof is also known as diagonalization .
In 1874 Georg Cantor found or published a proof of the countability of rational numbers and algebraic numbers by using the "first Cantor diagonal method". At the same time he published a proof of the uncountability of real numbers including the inference of the existence of non-algebraic real numbers. In the years 1890 and 1891 he found or published the proof that the power set of an arbitrary set is more powerful than this and that in particular the power set of natural numbers is uncountable. This proof is called "Second Cantor's Diagonal Method" and was the trigger for the establishment of the transfinite set theory by Georg Cantor in the years 1895 to 1897. The uncountability proofs also prove the uncountability of the continuum.