# Thread: how to prove this countability

1. ## how to prove this countability

2. Show $\mathbb{Z}[X_1,X_2,...,X_d]$ is countable.
Since by definition, $\mathbb{Z}[X_1,...,X_d] = \mathbb{Z}[X_1,...,X_{d-1}][X_d]$ it remains to prove that if $A$ is countable then $A[X]$ is countable.

Let $A_n[X] = \{ f(x) \in A[x] : \deg f(x) = n \}$ for $n\geq 0$.

Thus, $A[x] = \bigcup_{n=0}^{\infty} A_n[x]$.
Also, $|A_n[x]| = |A^n|$ is clear by defining a bijection.
Since each ( $n\geq 0$) $A^n$ is countable.
We have that $A[x]$ is a countable union of countable sets.
Thus, it is countable.

3. Show there are uncountably many transcendental numbers
.

Given an algebraic number $\alpha$ it needs to satisfy a non-zero polynomial over $\mathbb{Q}$. There are countably many non-zero polynomials over $\mathbb{Q}$ and each one has finitely many zeros. Since every algebraic number is among those polynomials and conversely a zero of one of those polynomials is algebraic it follows there are countably many algebraic numbers. Thus, there have to be uncountably many transcendental numbers. Because $|R| > |N|$.