Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers?
The question was: Let $C$ be a set of Cauchy sequences. $$\forall x \in \mathbb R:\exists(a_n)\in C:a_n\to x$$ The question was answered by user72694: Consider an infinite decimal expansion of $x$, and truncate it at rank $n$. This gives a sequence $(a_n)$ that tends to $x$.
To which I commented: It was mentioned to me that "Any Gödel-style enumeration can cover only those real numbers that are definable by formulas (in some fixed language). Thus, if in one's set theory, there are uncountably many real numbers, then some of these numbers must be undefinable (by formulas)". My comment is: A real number $x$ cannot therefore, in general, be given—and therefore cannot be truncated. This can only be done for a countable subset of the real numbers, such as $\mathbb Q$ (the rational numbers).
My question:
I am now reflecting on my comment above, whether or not it represents a constructivist view or not. Constructivists I gather are not following mainstream mathematics (Wikipedia: This viewpoint involves a verificational interpretation of the existence quantifier, which is at odds with its classical interpretation).
If I were to criticize the real numbers on the grounds that they cannot be defined—see my quotation above which is from Prof Karlis Podnieks by the way—I believe I would propose a constructivist view.
However, the answer, to which I made my comment, assumes that—or some arbitrary real $x$—can be given, i.e., constructed, so I tend to believe that contructivism is not necessarily involved here.
What do you think?