If I randomly told you something like, "I'll do that tomorrow", you would have absolutely no idea what I was talking about. But if I had said "I'm going to cut my hair. I'll do that tomorrow." you would have understood me completely.

Ok - so what's the point?

Well, what this simple example illustrates is: in order to communicate an idea in a clear and meaningful way, there must be some sort of groundwork that has been laid; some sort of assumption that allows us to get started on the same page.

This principle applies in any discursive setting, but it is particularly important in mathematics. A mathematician must be able to communicate in clear and meaningful ways not just to make his work easy to understand, but in fact to make his work MATHEMATICALLY valid! The language of mathematics must be finely tuned to allow only precise statements, and to exclude the possibility of ambiguity.

How have mathematicians overcome this obstacle? What have they done to insure the clarity and precision of their work?

They've created the field of Set Theory!

Broadly speaking, "Set Theory" is the foundation of much of modern mathematics. It allows mathematicians to quickly, easily and UNAMBIGUOUSLY define mathematical objects. Furthermore, it allows the relations between mathematical objects to be clearly and rigorously analyzed, due to a large amount of "structure", or "rules" which govern the behavior of the fundamental components of the theory.

"Set Theory" is not a "theory" in the common sense of the word. In mathematics, the term "Theory" is used when describing a subject within mathematics that has a distinct and consistent language, utilizes similar techniques, and involves similar concepts. So for example, one could refer to Arithmetic as "The Theory of Arithmetic", referring specifically to the well known language, concepts and technique of arithmetic. But just as arithmetic is not a "theory" in the common sense of the word, so too with Set Theory.

NAIVE VS. AXIOMATIC SET THEORY

Given the vast nature of the mathematical sciences, you might imagine that it has been difficult to establish a solid and encompassing foundation for the science - and you would be right! The development of set theory has been a long and challenging journey, starting with Georg Cantor and Richard Dedekind, and continuing to the present day. The first advances in the study of set theory relied largely on intuitive notions, and did not rely on formal axiomatic

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