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Why is cardinality of real number $2^{\aleph_0}$? While I know that real numbers can be constructed using sets of natural numbers, that solely does not mean that real number has cardinality $2^{\aleph_0}$. So, what makes real number cardinality $2^{\aleph_0}$? I can't find why it is like this in my textbook... |
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Use the Cantor–Bernstein–Schroeder theorem: Choose $x\in \mathbb{R}$, and let $\{b_i\}_{i=1}^{\infty}$ be the (shortest) binary expansion of $\frac{1}{2}+\frac{\arctan x}{\pi}$ (which is in $(0,1)$). Then define $\phi(x) = \{ i | b_i = 1 \}$. This establishes an injective map $\phi:\mathbb{R}\to 2^{\mathbb{N}}$. To go the other way, suppose $A \subset \mathbb{N}$. Then let $t_i = 1_A(i)$ (indicator function), and define $\eta(A) = \sum_{i=1}^{\infty}\frac{t_i}{3^i}$. This establishes an injective map $\eta : 2^{\mathbb{N}} \to \mathbb{R}$. The desired result follows from the Cantor–Bernstein–Schroeder theorem. |
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