Rational-number sets have same infinity level as counting-number sets, as proved by Cantor's diagonal process {diagonalization} {Cantor diagonal process} {diagonal proof}.
rational numbers
Make table with infinite rows and columns. Cells have positive rational numbers. First row is a positive-integer series. Second row is a positive-integer series, each divided by 2. nth row is a positive-integer series, each divided by n.
To count fractions, start at top left 0. Go down one row to 0/2. Go diagonally up and right to first row at 1. Go right one column to 2. Go diagonally down and left to first column at 0/3. Repeat to cover all cells and count all fractions. Fractions count only once, establishing one-to-one correspondence between counting numbers and rational numbers.
real numbers
The real-number set has higher infinity than counting-number set. List real numbers in sequence as table. Rows are real numbers. Columns are digits. Along diagonals through table are real numbers, with one digit from each row and column. Change all digits of main-diagonal real number. Resulting real number is not any real number already in table, because all row and column digits have changed. Therefore, real numbers number more than counting numbers, and no one-to-one correspondence exists between counting numbers and real numbers.
Mathematical Sciences>Set Theory>Infinite Set
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Date Modified: 2022.0224