I'm actually interested in what these results of practical significance would be. Practical, as in actually useful for technology or such.
I'm also not sure what you mean with "finitist framework". As I said, analysis existed before set theory, it doesn't require any special framework. In fact, the system of natural numbers, real numbers, and complex numbers can be axiomatized just with second-order logic. Without set theory, let alone a transfinite set theory like ZFC. And normal mathematicians wouldn't even use a formal logic here, they would just write down those axioms in plain English.
I'm also not sure what you mean with "finitist framework". As I said, analysis existed before set theory, it doesn't require any special framework. In fact, the system of natural numbers, real numbers, and complex numbers can be axiomatized just with second-order logic. Without set theory, let alone a transfinite set theory like ZFC. And normal mathematicians wouldn't even use a formal logic here, they would just write down those axioms in plain English.