Cantor's Diagonal Argument

We use Cantor's diagonal argument and Cantor's hypothesis to prove that there are more real numbers than natural numbers when considering comparison between natural numbers and radix of real sets. This article first outlines this discussion and continues to explain some of its effects. Next, I will consider his comments on Wittgenstein and Cantor's claims, that is, the overrun abstraction, indefinite use, and the assumption that all sets can be properly ordered.

The diagonal argument of Cantor indicates that the exponentiation set of a set always has a stricter radix than the set itself (regardless of whether it is infinite or not) (Informally, the exponentiation set is not larger than the original set It is not bad). In particular, the Cantor theorem shows that the power set of the countable infinite set is infinite. The exponentiation set of a set of natural numbers can correspond one-to-one with a set of real numbers (see Radix of continuum). The exponentiation set and union set of set S, intersection operation and complement operation can be thought of as examples of Boolean algebra prototypes. In fact, it can be shown that any finite Boolean algebra is isomorphic to boolean algebra of the power set of the finite set. In the case of infinite Boolean algebra, this is no longer true, but each infinite Boolean algebra can be represented as a partial algebra of power set Boolean algebra (see Stone's expression theorem).

The set of advance numbers is infinite infinite. Polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeros, the algebraic number must also be countable. However, Cantor 's diagonal argument proves that real numbers (thus multiple numbers) can not be counted. Real numbers are a combination of algebra and transcendental numbers, so you can not count them all. This makes it impossible to count a priori number