I think constructivists have to acknowledge algebraic numbers. In algebra you can construct speziell fields with them. This numbers are constructeble and you can create number systems with them. For example is the golden ratio such a number. You can use the golden ratio for a number system.
Transzendentale numbers like pi are not constructeble. So for a constructivist they don’t exist (also the Euler Number). It is indeed a philosophical question. But a think the main point for pi is that pi is not constructeble.
Mathematical constructivism is subdivided into several varieties each with their own set of rules. So in the post indentified mathematical constructivism with the more extreme variety. I didn't make that clear in the post sorry about that.
Furthermore, certain types of constructivism allow countable infinity so then you can already construct irrational numbers.
What is a Speziell field?
I see. I was talking about constructivism in a philosophical way. In my interpretation are only constructeble thinks are true.
Sorry for my bad english field are an algebraic structure. Are was talking about field extensions. You can add to rational numbers in an infinite way constructeble numbers and get algebraic numbers. https://en.wikipedia.org/wiki/Constructible_number
Ah yes so the quadratic closure of the rational numbers. So in this case you are assuming that countable infinity exists.