I've developed great interests in many areas related to pure and analitical sciences, especially physics, philosophy and math. This forum is a perfect place for me to learn some new materials. Since this is my first post in the forum, I'd like to introduce myself. I'm 17 year old studen from Canada.
So to the topic of set theory. I didn't get a chance ot read the whole of the arguments, but here is a fundamental problema friend of mine (also in high school) has raised. When numbers were invented by our ancestors, whether it is based on digits or otherwise, they were intuitively meant to be discrete. A man only had an integer amount of fingers. Well, that's a given. But when the Greeks finallized the concept of geometry which deals exclusively with space (which is assumed, axiomatically or not, to be continuous). In orde to give quantative measures of space, they employed the abstract concept of numbers. The mistake that the Greeks made was, in my opinion, that they employed a throughly discrete system of measurement to describe something that is by nature continuous. To the humble limits of my knowledge, no one bothered ask whether discrete systems are fit to describe continuity. A point has no dimension and thus, no length. A ray, however, has a one sided limit, but under our current mathematical systems there is no way to locate it if it is located at some real location. Hence, the concept of limits developed.
I hightly believe in the computational view of mathematics. It is not that continuity does not exist, but our number system, which is based on the repeation of natural numbers (think PI or), is unable to locate it precisely. The process of constructing real numbers , or realizing them, undoutedly arises from manipulating natural numbers. The golden ratio for example ((1+5^(1/2))/2). The adaptations of the natural numbers into the real numbers inevitably raises problems about continuity, and perhaps about infinity as well.
So in conclusion, there is a possibility that our number system might not be suitable to describe continuity without raising problems. Thank you for reading this. I hope I hadn't wasted you time.