In the Introduction of BookofProofs, we have learned about the axiomatic method, how it shaped mathematics over centuries and why it is important for education. In this part of BoP we will provide all tools necessary to provide a strict mathematical foundation of the axiomatic method itself.
It is important to notice, that we will be using two different levels of reasoning in order to do so: * the object level of the concepts we will introduce, and * the meta-level of mathematics necessary to properly define these concepts.
Particular caution is recommended not to mix up both levels of reasoning. While the meta-level provides us with all logical and mathematical tools and freedom necessary to prove and define the concepts of the object level, we do not have these tools and freedom at our disposal at the object level at all. At the object level, we will be introducing a logical calculus step by step, pretending that we even do not know what an alphabet is and trying to construct formal languages, their syntax, and semantics. We will learn at the object level from a strict and formal point of view, what axioms are, what are rules of inference, to derive logically theorems from axioms them and what can be said about the truth of these theorems. On the object level, we will be using the notion of a logical calculus to denote any formal system using the axiomatic method.
All these concepts sound highly sophisticated and discouraging for the novice. If you are a novice, be assured that they are not. In fact, they are quite straightforward and you are invited to read further.