I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
2.74 Definition Cantor set
The Cantor set $C$ is $[0,1]\setminus (\bigcup_{n=1}^{\infty} G_n)$, where $G_1=(\frac{1}{3},\frac{2}{3})$ and $G_n$ for $n>1$ is the union of the middle-third open intervals in the intervals of $[0,1]\setminus (\bigcup_{j=1}^{n-1} G_j).$
The author wrote as follows on p.$2010_3$ in this book:
2.75 base $3$ description of the Cantor set
The Cantor set $C$ is the set of numbers in $[0,1]$ that have a base $3$ representation containing only $0$s and $2$s.The two endpoints of each interval in each $G_n$ are in the Cantor set. However, many elements of the Cantor set are not endpoints of any interval in any $G_n.$
For example, Exercise 14 asks you to show that $\frac{1}{4}$ and $\frac{9}{13}$ are in the Cantor set; neither of those numbers is an endpoint of any interval in any $G_n.$ An example of an irrational number in the Cantor set is $\sum_{n=1}^\infty\frac{2}{3^{n!}}.$
It is unknown whether or not every number in the Cantor set is either rational or transcendental (meaning not the root of a polynomial with integer coefficients).
$\frac{1}{3}=0.02222\dots_3$ is a rational number in $C$.
$\sum_{n=1}^\infty\frac{2}{3^{n!}}$ is an irrational number in $C$.
My question is here:
The author wrote as follows:
It is unknown whether or not every number in the Cantor set is either rational or transcendental.
The author didn't write as follows:
It is unknown whether or not every number in the Cantor set is either rational or an algebraic number.
Does this mean implicitly mathematicians know there exists at least one transcendental number in $C$?
Does this mean implicitly mathematicians don't know if there is an algebraic number in $C$?
Best Answer
The algebraic numbers are countable. The Cantor set is not. Therefore, there is at least one transcendental number in the Cantor set. In fact, at least two distinct ones, in fact infinitely many. Actually, almost every number in the Cantor set is transcendetal.
What the author is saying, quite explicitly, is that there are some algebraic numbers in the Cantor set, but we do not know if there are any of them that are irrational as well (e.g. $\sqrt\frac12$ is an irrational algebraic number).