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May 28, 2023 · The following gives the original idea of Cantor’s proof. Cantor devised the following function f: [0, 1] × [0, 1] → [0, 1]. First, we represent the coordinates of any point (x, y) ∈ [0, 1] × [0, 1] by their decimal representations x = 0.a1a2a3... and y = 0.b1b2b3....
Dec 30, 2020 · According to his first assumption, Cantor thinks that he should not find any other number between (0–1). He also knows, however, that he must prove this mathematically. That is why he starts looking for a number that he thinks is not between (0–1), b .
Jun 10, 2019 · Cantor sets are sometimes referred to as the “Cantor comb” or the “no middle third set.” If you are unfamiliar with these terms, they will make more sense when we talk about constructing a set.
Many people believe that the result known as Cantor’s theorem says that the real numbers, \(\mathbb{R}\), have a greater cardinality than the natural numbers, \(\mathbb{N}\). That isn’t quite right. In fact, Cantor’s theorem is a much broader statement, one of whose consequences is that \(|\mathbb{R}| > |\mathbb{N}|\).
By taking the number produced by the diagonal argument you ensure at the $n$-th step that you are not any of $a_1,\ldots,a_n$. When the induction "ends" you have produced a real number which is none of the $a_i,\ i\in\mathbb N$.
Jun 29, 2023 · This chapter addresses Georg Cantor’s transfinite numbers, with some emphasis upon the philosophical questions they raise. We recall that Cantor’s attempts to justify transfinites were directed at mathematicians and philosophers alike—both ancient...
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Sep 22, 2017 · In trying to construct the Real Numbers , it is important to recognize that many times, real numbers are simply identified by an approximate rational number. Consider the real number =3.14169…. can be approximated by multiple rational number such as 3.14, 3.14159, or 3.1415926536.
