Natural Numbers

What is a natural number? What is a number? Such fundamental questions are hard to answer because you may stumble on fallacies like circularityUsing an advanced fact to prove a more elementary fact. Mathematically, we define natural numbers to be any number in this set:

$N = \{1,2,3,4,...\}$ Thus, we also refer to $N$ as the set of natural numbers. These numbers will serve as our first principles as we prove our way through mathematics into fields like linear algebra and calculus. If we take this notation for what it is, we are still left with many holes in our understanding. For example, when does this set end? This question and similar ones require a sharper toolset that what we currently have, so we move to our next step in mathematics: incrementation. In these notes, incrementation will be represented as ++, similar to modern coding languages like C++. A new way to define $N$ would be $N = \{0, 0++, (0++)++, ...\}$ With this understanding, natural numbers can be fully developed via the Peano Axioms, which fully flesh out this number system and lead us to first principled Addition