# [FOM] Who was the first to accept undefinable individuals in mathematics?

Sat Mar 14 12:12:22 EDT 2009

```While we may need Sturm's theorem to compute the algebraic numbers, we
do not need Sturm's theorem to define an enumeration of all algebraic
numbers. An algebraic number is defined by its equation
a_0x^n+a_1x^{n-1}+...+a_n=0, with integer a_i. We can enumerate all
tuples of integers (this is something Cantor knew how to do), and then
enumerate all the roots of each equation in the increasing order: the
smallest root of equation No. 1, the next smallest root of equation No.
1, ...,the smallest root of equation No. 2, ...

This description makes the enumeration of algebraic numbers perfectly
definable.

> From: William Tait

> A better example is from Cantor's original paper on the non-
> denumerability of the reals. He concluded that there is a
> transcendental number between a and b (when a<b) and made a point of
> the fact that this result is significant even if it does not yield a
> definition of such a transcendental. The transcendental can in fact be

> defined in terms of an enumeration of all the algebraic numbers
> between a and b; but in Cantor's time, before Sturm's Theorem, no such

> enumeration could be defined.

```