<Dd> The set T is uncountable . </Dd> <P> He assumes for contradiction that T was countable . Then all its elements could be written as an enumeration s, s,..., s,.... Applying the previous theorem to this enumeration would produce a sequence s not belonging to the enumeration . However, s was an element of T and should therefore be in the enumeration . This contradicts the original assumption, so T must be uncountable . </P> <P> The interpretation of Cantor's result will depend upon one's view of mathematics . To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibility that the latter are subcountable . In the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers . </P> <P> The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from the above result . To prove this, an injection will be constructed from the set T of infinite binary strings to the set R of real numbers . Since T is uncountable, the image of this function, which is a subset of R, is uncountable . Therefore, R is uncountable . Also, by using a method of construction devised by Cantor, a bijection will be constructed between T and R. Therefore, T and R have the same cardinality, which is called the "cardinality of the continuum" and is usually denoted by c (\ displaystyle (\ mathfrak (c))) or 2 א 0 (\ displaystyle 2 ^ (\ aleph _ (0))). </P>

Why is the set of real numbers uncountable