Cantor showed that the set of algebraic numbers is countable and the set of transcendental numbers is uncountable. Is there any (modern)book with the proof of these theorems?
2026-05-05 01:15:06.1777943706
Reference request: modern reference for Cantor's theorems of size of algebraic and transcendental numbers?
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The fact that there are only countably many algebraic numbers appears as a theorem in Mathematical Analysis I by Zorich. Then the existence of transcendental numbers is stated as a corollary of the theorem that $\mathbf{R}$ is uncountable.
The fact that the set of transcendental numbers has the cardinality of the continuum is only an exercise there.
However, the fact that there are uncountably many transcendental numbers is a direct consequence of the facts I mentioned that appear as theorems together with the theorem that the union of two countable sets is also countable.