About $\blacktriangleright$ Specifying syntax An apple pie from scratch Perspective

### Meta

This entry explains the motivation and content for this website, the notation for the wiki and the structure of the graph. All of the content can viewed via an interactive graph, which one can also access from each entry by clicking the yellow lemon in the upper right corner.

#### Motivation

The motivation to for this project initially came from my wish to formally write down theories of physics and put them in relationship to each other. Put in the framework of a graph, this should immediately answer questions like “What are the different field theories?”, “Which of those can be viewed as special cases of others?”, “How exactly does Newtonian gravity relate to special relativity?” Stuff like this.

Soon I became aware that this requires a lot of compatible mathematical language and one ends up writing more on mathematical than physical concepts. And more than before, had to take a bite of sour fruit of lacking mathematical background. I decided to learn some math from scratch, connecting my fuzzy high end concepts in my head to solid predicate logic. So the current incarnation of my graph idea takes the form of interconnected notebook with emphasis on mathematics itself. This page gives an overview of the chosen approach.

#### Content

The basic structure of the typical entries is this: First I have a Definition section, were a concept is defined. Then follows a Discussion section where I write down elaborations and personal insights I want to document. This is then followed by the Parents section, which defines the connections to other entries. (The entries at the root of the graph are written in a more explanatory style than the rest of the entries and they have no clear Definition section. The current page is a good example of this.)

The edges in the graph represent relations such as “is subset of”, “is requirement for the present definition”. Every entry should reference at least one collection of other entries, which if taken as an oracle, contain all the information to define it. That rule is followed to a “reasonable” extent. There is no point to link to “set union” or “exponential function” to every single definition which uses these. If I feel it's worth it, other possible definitions are added in the Discussion section of an entry. Note that since I don't want to prohibit myself from re-writing entries, there is no control mechanism which prevents the main definitions of individual entries to end up being presented in a circular way. So in case that should happen, and in case one is interested in a linear hierarchy, one must check an alternative definition to break the circle. While many entries suggest an chronology of definitions, from the above it's clear that graph does not represent one long chain of derivations from any particular set of axioms. The spirit of the graph is that of impredicative definitions, but I don't want to enforce any such approach for the same of it.

The first few entries are also naturally concerned with foundational issues. So for example, the entry Set theory lists possible set theory axioms. After this, most of the following entries define mathematical concepts in terms of sets. But I also want to make the point that the content of this wiki is not to be taken with respect to any specific set theory, or any other fixed axiomatic framework for that matter. Some definitions are even presented in a purely category theoretical language.

Add info on Wat and An apple pie from scratch.

#### Guide line

It's a tricky issue, but I (sometimes) think each (main) definition should be as standalone as possible, only stating the definition in terms of other stuff in the “Alternative definitions” block
nevertheless, of course the objects are still ordered by subset inclusion.
Most definitions are sets in ZFC and classical on purpose. The discussion section might deal with alternative deifnitions and “more modern” elaborations of the concept.
The idea to hold the definition classical is to make me able to read the always mostly classical literature. Also, if I take a non-classical path, then I'd have to make a judgment which approach is good, and I might end up wanting to rewrite everything again and again. I don't want to get into this.

#### Practical issues

todo: make a single “Formalities” section where you list all the issues.

This wiki is pretty rigorous, but there are some standard conventions I have to adopt. For example: I will not bother pointing out if the plus sign “$+$” I'm using is that for real or that for complex numbers. And I will not care if I formally defined the triple $\langle a,b,c\rangle$ as $\langle \langle a,b\rangle,c\rangle$ or $\langle a,\langle b,c\rangle\rangle$. I elaborate on structural issues some more in the Discussion section below.

##### Set theory vs. structure

I'm personally not even a big fan of sets. E.g. the Cauchy construction of the reals as equivalence classes of infinite sequences (i.e. each number is a gigantic set) is completely ridiculous if taken at face value. But that overhead is just a side product and th point of using set theoretical models of mathematical objects is really that we can use a single language and a plus is also that majority of the literature is written in that language. I write down certain set definitions which fulfill the defining property of the objects we are interested and then try to only work with those.

With the standard approach of encoding mathematical concepts in terms of sets come some issues which might be considered not only unaesthetic but also unpractical. Here two examples:

1. Adopt the standard definitions for the ordered pair and the natural numbers: The ordered pair $\langle a,b\rangle$ of two sets $a$ and $b$ is identified with the set $\{\{a\},\{a,b\}\}$, because in this way one can distinguish between the first and second element. The number zero $0$ is identified with the empty set $\emptyset$ and the number $1$ is the set which contains the number zero, i.e. $\{\emptyset\}$), because then there is a way to represent addition via set union. Now that is a standard approach, but it also implies $1\in\langle 0, 7\rangle$ is true, which is obviously merely an artifact of our modeling. That statement is true for the ordered pair defined as above, but there are other models of “the theory of the orders pair” in which it's not. (I.e. the logical axioms together with the defining property $(\langle a,b\rangle = \langle x,y\rangle)\Leftrightarrow (a=x\land b=y)$ taken as an axiom)

2. You can go and define the collection of real numbers $\{1,2.3,2\pi,\dots\}$ as some set $\mathbb R$ with the structure of addition $+$ and multiplication defined on them. Now if you model the set of complex numbers $\mathbb C$, you must make a decision: Do you want the real numbers above to be direct elements of it, or do you rather want to model all your complex number in the same fashion? For example as $a+ib\equiv\langle a,b\rangle$, with real and imaginary part stored in an ordered pair? In principle, the truth of $2\pi\in \mathbb C$ depends on it. Of course, the structure including $2\pi$ and the one including $\langle 2\pi,0\rangle$ are perfectly isomorphic.

Usually, one never sees people care about these issues as the detailed way in which the mathematical concepts are modeled by a set theory is often not of concern. The category theoretical approach tries to capture structural aspects more directly. Everything is defined only up to isomorphisms - with the drawback of making some discussions more abstract.

##### Notation

For the notation used in the first few entries, see the relevant section in Glossary.