TITLE: Is the analysis as taught in universities in fact the analysis of definable numbers?
QUESTION [109 upvotes]: Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:

All numbers are divided into two classes: those which can be unambiguously defined by a limited set of their properties (definable) and those such that for any limited set of their properties there is at least one other number which also satisfies all these properties (undefinable).
It is evident that since the number of properties is countable, the set of definable numbers is countable. So the set of undefinable numbers forms a continuum.
It is impossible to give an example of an undefinable number and one researcher cannot communicate an undefinable number to the other. Whatever number of properties he communicates there is always another number which satisfies all these properties so the researchers cannot be confident whether they are speaking about the same number.
However there are probability based algorithms which give an undefinable number as a limit, for example, by throwing dice and writing consecutive numbers after the decimal point.

But the main question that bothered me was that the analysis course we received heavily relied on constructs such as "let $a$ be a number such that...", "for each $s$ in the interval..." etc. These seemed to heavily exploit the properties of definable numbers and as such one can expect the theorems of analysis to be correct only on the set of definable numbers. Even the definitions of arithmetic operations over reals assumed the numbers are definable. Unfortunately one cannot take an undefinable number to bring a counter-example just because there is no example of undefinable number. How can we know that all of those theorems of analysis are true for the whole continuum and not just for a countable subset?

REPLY [16 votes]: You can also talk about arithmetically definable real numbers:  those for which the Dedekind cut of rationals is of the form:
$$\{m/n: \forall x_1 \exists x_2 \ldots \forall x_{k-1} \exists x_k\,
p(m,n,x_1,\ldots,x_k)=0\},$$
where the $x$'s range over integers, and $p$ is a polynomial with integer coefficients.  
Then on this definition of definability:  $e$ and $\pi$ and all the familiar reals are definable.  Only countably many numbers are definable.  There must be other real numbers which are undefinable.  And it all makes sense, and is even provably consistent, in ordinary set theory.
A standard reference for this way of thinking is the system $ACA_0$ in Simpson's Subsystems of Second-Order Arithmetic.  
The cost of this metamathematical simplicity is a small change to the mathematics:  any definable bounded sequence of reals has a definable least upper bound, but an uncountable definable set of reals may not.  Feferman's notes on Predicative Foundations of Analysis show how to develop standard analysis on this basis.  If we changed mathematics as taught in universities to be based on predicative analysis, few undergraduates or people outside the math department would notice much difference.