Cardinality of the set of algebraically independent transcendental numbers

Question

It is not difficult to see that there must be an uncountable infinity of transcendental numbers – but has anybody yet proved that there is an uncountable infinity of algebraically independent transcendental numbers ?

Answer 1

I assume we are talking about $\mathbb{R}$ or $\mathbb{C}$ as an extension of $\mathbb{Q}$. The transcendance degree of an extension is defined to be the cardinality of the maximal set of algebraically independant numbers. Zorn lemma ensures that this maximal set always exist i.e for any field extension ; it also allows to prove that two such maximal sets have the same cardinality. In the case of $\mathbb{R}/\mathbb{Q}$ (or $\mathbb{C}/\mathbb{Q}$) this maximal set is uncountable. To prove it suffices to bear in mind that above any $\alpha$ transcendental over $\mathbb{Q}$ one can find countable many $\mathbb{Q}$ algebraically dependant numbers, simply because $\mathbb{Q}$ is countable.

Answer 2

Suppose there are countably many transcendental numbers which can be used algebraically to construct all real numbers. All such expressions representing reals can be written in some specified way with a finite alphabet (write polynomials in the usual notation with constants having integral subscripts representing algebraically independent transcendental numbers). There are only countably many finite expressions in a finite language. So then a countable number of expressions maps onto the reals, and thus the reals are countable.