Introduction to Codex

Codex is my personal math journal.

Structure

In Codex, pages are organized according to their branches (e.g. Set Theory, Analysis) and topics (e.g. Functions, Series). Each page discusses a specific concept within a topic (e.g. Inverse Function from the topic Functions).

Codex is mostly made of entries. They look like this:

XXX-YYY-ZZZ.

Blablablabla…

Each entry is either an axiom, a definition, a proposition or a remark (called the entry’s type), labeled in the following format:

<entry type>-<concept>(-<tag1>-<tag2>-<...>)

• All labels contain only uppercase letters, (maybe) numbers, and dashes.

• <entry type> is the abbreviation of the entry’s type:

  • Axiom 🠆 AX
  • Definition 🠆 DEF
  • Proposition 🠆 PROP
  • Remark 🠆 REM

  In Codex, proposition is a generalized term for mathematical results (theorems, lemmas, etc.). Every result, regardless of its role or importance, is labeled as a proposition.

• <concept> is the abbreviation for the discussed concept on a page (e.g. Subset 🠆 SUB). It is also the code name of the page. The choice of this abbreviation is arbitrary.

• <tag1>-<tag2>-<...> are optional tags used to further identify the entry. Usually only one tag is needed. Sometimes there is no tag at all, especially when the entry is the primary definition of a concept (e.g. DEF-SUB is the definition of subset). The choice of these tags is arbitrary.

Why?

Why not use the conventional format (“Definition 3.12”, “Theorem 5.4”)?

The more familiar system, which labels entries based on the order of their occurrences, works well for static texts. However, Codex is dynamic by its nature, meaning that entries are expected to be added, removed and rearranged every now and then. With the conventional system, such changes would require a lot of renumbering.

Using individual, symbolic labels avoids the problem, as the order doesn’t matter. Another advantage is that they are more descriptive, making them easier to remember (“Definition 5.1” vs. “DEF-SUB”).

Conventions

• In logical formulas, colons split quantifiers and statements:

  \[\forall a\,\exists b : \varphi(a,b).\]
  • It reduces the need for parentheses.

• In logical formulas, $\rimp$ is used for material implication instead of $\Rightarrow$.

  • $\Rightarrow$ is reserved for implication in the meta langauge, i.e. as a shorthand for words like so that, it follows that, therefore in the argumentation. Example: If $x\in\R$, then

    \[x^2\geq 0 \quad\Rightarrow\quad x^2+9\geq 9 \quad\Rightarrow\quad \sqrt{x^2+9}\geq 3.\]

    Meanwhile, $\rimp$ is a binary operator in logic, just like $\land$ and $\lor$.

  • The same applies for $\leftrightarrow$.

• $\subseteq$ for “subset” and $\subset$ for “proper subset”, instead of $\subset$ and $\subsetneq$.

  • $(\subseteq,\subset)$ is analogous to $(\leq,<)$.
  • $(\subseteq,\subset)$ is in my opinion more elegant.

• $\{x:\varphi(x)\}$ instead of $\{x\mid\varphi(x)\}$ for set-builders.

  • $\{x:\varphi(x)\}$ reads more naturally to me, since the colon itself carries the meaning of such that.
  • Consistent with the use of colons in logical formulas.

• $R[A]$ for the image of $A$ under $R$.

  • So that the image of a function, $f[A]$, is clearly distinguished from the value $f(A)$.

• $R^{-1}$ for both converse relation and inverse function.

  • The inverse function of an injection $f$ is precisely the converse relation of $f$.

• $X\equ Y$ for “$X$ and $Y$ are equinumerous”;
  $X\lequ Y$ for “$X$ is not greater than $Y$”;
  $X\lnequ Y$ for “$X$ is strictly smaller than $Y$”.

  • $(\equ,\lequ)$ is analogous to $(=,\leq)$.
  • $\lnequ$ instead of $<$ to avoid ambiguity.
  • $\sim$ is reserved for denoting a general relation.

• $\N$ for the set of natural numbers, including $0$.

  • $\N=\omega$ is defined as an ordinal, so it naturally contains $0$.

The following notations and definitions are either made up or less common:

• The set of all $\kappa$-sized subsets of $X$ is denoted by $\powerset_\kappa(X)$.

• The equivalence class of $a$ with respect to $\sim$ is denoted by $[a]_{\sim}$.

• The set of all functions from $X$ to $Y$ is usually denoted by $\fun(X,Y)$.

• That $a$ and $b$ are conjugate, is written

  \[a \stackrel{\conj}{\sim} b.\]

• Uniquely unital magma: A magma that has exactly one neutral element.

• Uniquely invertible magma: A uniquely unital magma where every element has exactly one inverse.

Other Notes

• By default, we work in $\ZF$ or $\ZFC$.

• [$\limp\AC$] means a result or a definition relies on $\AC$.

  [$\rimp\AC$] means a result implies $\AC$.

  [$\lrimp\AC$] means a result is equivalent to $\AC$.