\ch{Proofs}

\s{Peano Axioms}

This section covers the Peano axioms. As I said in \cref{s: Motivation}, these
are a way for mathematicians to understand arithmetic.

Arithmetic (hopefully) seems simple enough, and easy to understand. Maybe an
expression like

\[ (2048282 \times 33221) + (3254 \times 11) \]

seems difficult to calculate, but you hopefully understand what each of the
operators mean in concept. If you don't, well. In theory, reading this chapter
alone will teach you arithmetic. However, I wrote this chapter assuming you
already know arithmetic.

So, why is it important that you read this chapter?

Arithmetic is pretty simple and easy to understand. However, later on in this
book, we're going to approach concepts that aren't so simple and easy to
understand. Mathematicians have a systemic approach to these problems. This
approach is called ``mathematical proof.'' We prove things
mathematically. Instead of approaching new concepts with proofs, I'm instead
going to use proofs to illustrate some (hopefully) familiar concepts.

Alright, with all that out of the way, let's get started.

The basic idea of proofs is, you take a small set of obvious facts, called {\it
  axioms}, chain them together to make {\it theorems}. The following obvious
facts, or axioms, are called the ``Peano Axioms.'' They describe what we call
``natural numbers.'' Natural numbers are the numbers
$\{0, 1, 2, 3, 4, 5, 6, \ldots\}$.

%  I got these from Landau's book, but there's nowhere where it isn't awkward to
%  cite him. So, instead, I'll use \nocite

\nocite{landau-analysis}

\begin{description}
\item[Axiom 1] $0$ is a natural number. Again, obvious.

  I'm going to use letters in the place of numbers right here. So, if I say
  ``$x$ is a natural number,'' that means that $x$ is a placeholder for one of
  the numbers in $\{0,1,2,3,4,5,6,\ldots\}$. I could use any letter, such as
  $a$, $b$, $q$, $r$, $\theta$, $\Gamma$, or $\aleph$. If I use a letter instead
  of a number, it usually means either

  \begin{enumerate}
  \item it doesn't matter which number I choose, or
  \item it does matter which number I choose, but I don't know which number it
    is yet.
  \end{enumerate}

\item[Axiom 2] If $x$ is a natural number, it is true that $x \equiv x$.

  You can read that $\equiv$ sign as $=$, for the time being. There are some
  subtle differences between the two, which I will get to in \cref{ch:
    Functions}. You are supposed to read $a \equiv b$ as one of these:

  \begin{enumerate}
  \item ``$a$ is equivalent to $b$,''
  \item ``$a$ is identically equivalent to $b$,''
  \item ``$a$ is congruent to $b$.''
  \end{enumerate}

  $=$ should be read as ``$a$ is equal to $b$,'' or ``$a$ equals $b$.'' Again
  the difference between $\equiv$ and $=$ isn't really important until \cref{ch:
    Functions}.

  If you don't know what either of those signs are, $a \equiv b$ or $a = b$
  means ``$a$ is the same thing as $b$.'' The difference can be summarized as
  $= = \equiv \not \equiv = $.
  
  So, in essence, this axiom says that each number is the same thing as
  itself. This is hopefully very obvious.

  A math person would state this axiom as ``congruence is reflexive.''
  
\item[Axiom 3] If $x$ and $y$ are both natural numbers, and $x \equiv y$, then
  it's true that $y \equiv x$. You can phrase this axiom as ``if two numbers are
  the same number, then they are the same number.'' A math person would state
  this axiom as ``congruence is symmetric.''

\item[Axiom 4] If $x$, $y$, and $z$ are all natural numbers, and $x \equiv y$,
  and $y \equiv z$ then it's true that $x \equiv z$. You can phrase this axiom
  as ``if three numbers are all the same number, then they are the same
  number.'' A math person would state this axiom as ``congruence is
  transitive.''
  
  These last three axioms mean that we can be lazy, and write things like
  $a \equiv b \equiv c \equiv a$.
  
\item[Axiom 5] If $x$ is a natural number, and we know $x \equiv y$, then it's
  also true that $y$ is a natural number. A math person would say ``congruence
  forms a closure.''
  
\item[Axiom 6] If $x$ is a natural number, then there is another number,
  $\suc(x)$, which is also a natural number. $\suc$ is short for ``successor.''
  You should read $\suc(x)$ as ``the successor of $x$.''  You can think of the
  successor as ``the next number.'' So, $\suc(0) \equiv 1$, $\suc(1) \equiv 2$,
  and so on.
  
\item[Axiom 7] There isn't a number whose successor is $0$. Basically this means
  ``$0$ is the lowest natural number.''
  
\item[Axiom 8] If $x$ and $y$ are both natural numbers, and we know
  $\suc(x) \equiv \suc(y)$, then it's true that $x \equiv y$. This is what we
  would call the ``converse'' of Axiom 6. That is, Axiom 6 tells us that we can
  always go ``up'' a number. This axiom (almost) tells us that we can go
  ``down'' a number. Axiom 7 defines the limit of this, meaning that $0$ is the
  only number where you can't go down any further.
  
  Now, the previous 8 axioms have basically said ``these numbers are all natural
  numbers.'' This next, and final axiom states ``these numbers are all {\it of}
  the natural numbers.''
  
\item[Axiom 9] Let's say $K$ is a set of numbers (a bunch of numbers). If we
  know that
  \begin{enumerate}
  \item if $0$ is in $K$, and
  \item if some number $x$ is in $K$, then $\suc(x)$ is in $K$,
  \item then $K$ contains every single natural number.
  \end{enumerate}

\end{description}

\pg{Continue with...}

\begin{enumerate}
\item $\N$ is a Monoid
\item Quasigroups
\item Loops
\item Groups
\item Abelian Groups
  \begin{enumerate}
   \item $\Z$
    \begin{enumerate}
    \item $\Z$ is an Abelian Group
    \item $\Z$ is a ring
    \end{enumerate}
  \end{enumerate}
\item Fields
  \begin{enumerate}
  \item $\R$
    \begin{enumerate}
    \item $\R$ is a field
    \end{enumerate}
  \end{enumerate}
\item Categories
\item Groupoids
\end{enumerate}