在实数中，Cantor 的对角线法说明“可数”不再成立。若假设所有实数都可列成与自然数的一一列表，构造一个新数：取列表中第 \(n\) 个实数的第 \(n\) 位并加1作为新数的第 \(n\) 位。这样得到的实数在第1位与第1个数不同、第2位与第2个数不同，依此类推，最终与列表中任何已有数都不同。于是该列表不可能包含所有实数，原假设被否定。结论是实数集的基数严格大于自然数集基数，且为不可数无限（uncountably infinite）。
在后续比较中，(0,1) 与 (0,2) 仍展现反直觉：虽然直觉上前者应是后者的一半，但映射 \(x \mapsto 2x\) 给出从前者到后者的双射，说明两者等势，基数相同。进一步，(0,1) 与全体实数 \(\mathbb{R}\) 也同阶；实数线上的任何非零长度区间在基数意义上都不更小。Cantor 的“天堂”表明：无限不只有一个量级，可数与不可数只是起点，存在更高层级；这些结果重塑了数学对“无穷大小”与“可定义性边界”的理解。
In the first two cases, size is judged by one-to-one correspondence. The natural numbers and even numbers can be perfectly paired by the map \(n \leftrightarrow 2n\), so although evens seem sparser, the two sets have the same cardinality and are both countably infinite. Then the rationals are arranged in a grid: row 1 is \(1/1,2/1,3/1,\dots\), row 2 is \(1/2,2/2,3/2,\dots\), and so on indefinitely. Scanning row by row never exhausts all rows, but a snaking (diagonal) traversal avoids this and, after skipping duplicates (such as \(2/2=1/1\)), maps each natural number to a distinct rational, proving the rationals are also countably infinite.
In the real numbers, Cantor’s diagonal argument shows “countable” fails. If we assume all real numbers can be listed in one-to-one correspondence with naturals, construct a new number: take the nth digit of the nth listed real and add 1 as the nth digit of the new number. The new number then differs from the first listed number in digit 1, from the second in digit 2, and so on, so it differs from every listed number. Hence the original list cannot contain all reals and the assumption is false. Therefore the cardinality of reals is strictly larger than naturals, and is uncountably infinite.
The comparison of intervals gives another counterintuitive result: although \((0,1)\) seems half the size of \((0,2)\), the map \(x\mapsto 2x\) gives a bijection, so they are equal in cardinality. In fact, \((0,1)\) is also the same size as the entire set of reals \(\mathbb{R}\), meaning any nonempty real interval has the same cardinality as the whole line. Cantor’s “paradise” shows that “countable” and “uncountable” are only initial levels of infinity, and there are infinitely many larger magnitudes. These ideas challenged mathematical intuition and reshaped the perceived limits of what mathematics can define and prove.