positive(?) definition of a transcendental number - as opposed to negative def. not an algebraic number

Question

Does this make sense? What's the formal definition of a transcendental number, but without saying it's not an algebraic number?

Answer 1

There is no other definition.

By analogy, there is essentially no definition of "irrational" other than "not rational".

There is a philosophical point here. Any theorem that says that an assertion about $a$ is equivalent to $a$ being rational could be taken as the definition. So, for example, you could define "a is irrational" as "the continued fraction for $a$ never terminates".

If you had an analogous theorem about transcendentals you could use it as the definition. But that would probably not be clearer or better pedagogically.

Answer 2

Here is the usual definition, rewritten to seem positive. For any nonzero polynomial $P$ with integer coefficients, denote $A_P:= \{x \in {\mathbb R}: P(x) \ne 0\}$. Then $\xi$ is transcedental iff it is in the intersection $$\cap_{\{P \in{\mathbb Z}[x]\setminus\{0\}\}} \, A_{P} \,.$$