There are models of the teams that are countable when looking at the model externally within ZFC. However, that model thinks it is uncountable. When you say “real numbers” what exactly do you refer to? I know you mean the standard model but other models think they are real numbers too.

How do you know a better understanding of these kinds of philosophical classifications will lead to a better grasp of the connections between mathematics and the “real” world? What is the definition of real world that excluded mathematics?