Uncountable amount

A set is called uncountable if it cannot be counted . A set is called countable if it is either finite or a bijection to the set of natural numbers exists. So a set is uncountable if its thickness (corresponds to the number of elements in finite sets) is greater than that of the set of natural numbers.

To put it clearly, a set is uncountable if every list of elements in the set is incomplete. ${\displaystyle x_{1},x_{2},x_{3},\ldots }$

Proof of the uncountability of real numbers

Cantor's second diagonal argument is a contradiction proof, with which he proved the uncountability of real numbers in 1877 . (The first diagonal argument is the proof of the countability of the rational numbers .)

Contrary to popular belief, this proof is not Cantor's first proof of the uncountability of real numbers. Cantor's first proof of uncountability was published in 1874, three years before his second diagonal argument. The first proof works with other properties of the real numbers and gets by without a number system .

Comparison of the power of a set and its power set

With a generalization of Cantor's method one can show that the set of all subsets of a set , the so-called power set of, is uncountable if it has infinitely many elements. More precisely: You can show that it has a greater thickness than . With the help of the power set, an infinite number of different classes of infinity can be constructed. ${\displaystyle M}$ ${\displaystyle P(M)}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle P(M)}$ ${\displaystyle M}$

The continuum hypothesis postulates that there are no uncountable sets whose thickness is smaller than that of real numbers. However, it has been shown that the continuum hypothesis can neither be proven nor refuted assuming the usual axioms .

literature

• Georg Cantor: On an elementary question in the theory of manifolds . In: German Mathematicians Association (ed.): Annual report of the German Mathematicians Association . tape 1 . Reimer, 1892, ISSN  0012-0456 , p. 75-78 ( uni-goettingen.de ).
• About a property of the epitome of all real algebraic numbers. (PDF) In: Journal for Pure and Applied Mathematics. 77, pp. 258-262.