Introduction
The natural numbers are perhaps one of the most (if not the most) fundamental mathematical objects. This is the case in at least two senses:
- For one thing, they are likely the oldest mathematical ideas that humans first became aware of and used for counting/tallying. In fact, even other, less intelligent animals are known to count to limited extents; thus, the origins of counting probably precede even human evolution itself. For more concrete evidence pertaining directly to human use of natural numbers, one can look at prehistoric artifacts such as the Ishango bone from the Stone Ages; it is conjectured to have been used to count lunar cycles.
- More crucially, for the purposes of this GitBook, the natural numbers appear everywhere in mathematics. They appear either explicitly as the direct objects of study themselves as in, say, arithmetic and number theory, or implicitly as necessary ingredients for defining other notions/structures such as cardinality, finiteness/infiniteness, integers, sequences, real numbers, etc.
In this GitBook I rigorously prove properties about the natural numbers from first principles. By first principles, I refer to basic logic, informal set theory and the Peano axioms. To understand the material in this book well, it is best if the reader has prior working knowledge of the first two principles. In other words, the ideal reader should be comfortable with mathematical proofs, understand what sets are (at least informally), understand what subsets, unions, intersections and power sets are, understand set functions and their associated concepts etc. I do provide a refresher on some of these preliminary materials in the first chapter of this book. However, my main material begins in the second chapter with the last of these first principles — the Peano axioms — which finally define the behavior of the natural numbers, the subject-matter of this book.