Feigenbaum's constant is a number like Pi or e. It is transcendental. Even though you can't write it down with a finite number of digits, you can write down a formula to compute as many digits as you want (if you are ready to boil the oceans, build a Dyson sphere, harness a Quasar) You can give it a name and refer to it directly.

Any formula like that provides a set of brackets, "real" numbers with a finite number of digits (e.g. names) that we can say that the "phony" number is between. We can make the brackets finer and finer, but you can't pick out one in particular.

Thus 3, pi, pi/e + 6, sqrt(pi-e) are more "real" than the the continuum we imagine between them. Being able to name things, for instance, makes it possible to talk about them.

Constructivist mathematics [1] is an approach to math that restricts the universe of discourse to objects that can be explicitly defined. With this restriction, the subset of the real numbers considered consists of the reals that are definable [2], such as those that are computable [3] and constructible [4].

Similarly, the axiom of choice allows for the existence of nondefinable choice functions [5] in certain cases, so is rejected.

Regarding the part about analysis, the field of computable analysis [6] exists to establish analysis on constructivist footing.

[1] https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

[2] https://en.wikipedia.org/wiki/Definable_real_number

[3] https://en.wikipedia.org/wiki/Computable_number

[4] https://en.wikipedia.org/wiki/Constructible_number

[5] https://en.wikipedia.org/wiki/Choice_function

[6] https://en.wikipedia.org/wiki/Computable_analysis