Uncountability in intuitionistic logic
I tried to address the questions you raise in &quot;Five stages of accepting constructive mathematics&quot;.
And here are some textbooks:

Constructive analysis by D. Bridges and E. Bishop is the &quot;bible&quot; of constructive mathematics.
Varieties of constructive mathematics by D. Bridges and F. Richman considers several varieties of constructive mathematics, and discusses some of the pathologies and niceties that they possess.
Constructivism in mathematics (volumes 1 and 2) by D. van Dalen and A. Troelstra provides an in-depth treatment of constructive mathematics.