Why did Gödel name his constructible universe $L$? It seems like Gödel didn't use the letter $L$ for his model before his book "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory", which is probably the first place it got used.
Do anyone of you know why he used the letter $L$? It does seem like a bit ad hoc in the book, where he names some functions $J_i$, then some other functions $K_i$ and then ends up defining $L$. But why $J_i$ then?
(I'm sorry if this is not suited for MO)
 A: So far as I know, there are four interpretations why Gödel uses $\mathrm{L}$ to denote the constructible universe.
(1) By Paul Bernays (see page 64 in Halbeisen 2017), Gödel originally used the old German script  to denote the constructible universe, where  is a capital $\mathrm{C}$. Although the appearance of  is very like the German script $\mathfrak{L}$ for $\mathrm{L}$, in fact it's one kind of the German script $\mathfrak{C}$ for $\mathrm{C}$. Furthermore Bernay's words don't show why Gödel uses $\mathrm{L}$ to denote constructible universe in 1940 (in fact he had introduced the constructible universe in 1938 which is a proceeding paper). So I don't think Bernays' interpretation is suitable (as desired).
(2) By Kai Hauser, the letter $\mathrm{L}$ comes from "law" since the model is constructed using some laws. But in 1940, Gödel didn't mention any law and the word "law" at all. Furthermore, if so, any other model should be named $\mathrm{L}$ since they are also constructed using some other laws. So I don't think Hauser's interpretation is suitable either.
(3) By someone (I don't remember who said this first), since von Neumann's universe is the first model of $\mathsf{ZFC}$ which is denoted by $\mathrm{V}$ and the constructible universe may be the second model of $\mathsf{ZFC}$, to distinguish from and show relations to $\mathrm{V}$, Gödel extends the angle of $\mathrm{V}$ to 90，rotates it down to the horizontal line, and then makes it become $\mathrm{L}$. Of course, it is a joke, and can't be suitable either.
(4) In my opinion, the only suitable interpretation for Gödel's using $\mathrm{L}$ to denote the constructible universe is that $\mathrm{L}$ is after $\mathrm{J,K}$ in order since Gödel defines the constructible universe $\mathrm{L}$ after defining some functions $\mathrm{J}_\mathrm{i}$ and $\mathrm{K}_\mathrm{i}$ for which beginning with $\mathrm{J}$ is possibly because almost all the letters before $\mathrm{J}$ have been used before introducing the constructible universe in 1940 (and in particular he uses $\mathfrak{Cls}(\mathrm{A})$ related to $\mathfrak{C}$ for $\mathrm{C}$ to denote that $\mathrm{A}$ is a class and $\mathrm{Y}\,\mathfrak{Con}\,\mathrm{X}$ to denote that $\mathrm{Y}$ is connex in $\mathrm{X}$). In fact in 1940, Gödel's usages of many symbols are very random, for example he uses $\mathfrak{A}$ to denote some operation and $\mathfrak{B}$ to denote some notion without any sufficient considerations.

By the way, there are also four interpretations why people use $\mathrm{V}$ to denote the cumulative universe of all sets, or the class of all sets.
(1) Peano used uses $\mathrm{V}$ for "verum", i.e., the propositional truth constant for true, and its upside down form $\Lambda$ for "falsum", i.e., the dual propositional false constant, on page VIII in 1889. He furthermore uses $\mathrm{V}$ to denote the class of all individuals, and $\Lambda$ to denote the class containing no individuals on page XI in 1889 in Latin. The following words improved from comments of @EmilJeřábek may show why Peano made such uses.

On page VIII, Peano uses $\mathrm{V}$ for "verum", i.e., the propositional truth constant for true (that would nowadays be denoted $\top$ or 1), and its upside down form $\Lambda$ for "falsum", i.e., the dual propositional false constant (that would nowadays be denoted $\bot$ or 0), and also, $-$, $\cap$, $\cup$ (that would nowadays be denoted  $\neg$, $\wedge$, $\vee$) for the propositional connectives. Then, on pages X–XI, he uses the same symbols $-$, $\cap$, $\cup$ to denote the corresponding operations on classes, and thus $\mathrm{V}$, $\Lambda$ becomes respectively the class of all individuals and the class containing no individuals.

(2) Whitehead and Rusell used $\mathrm{V}$ to denote the class of all objects, which is defined as $\mathrm{V}=\{x\mid x=x\}$ in modern notations, on page 229 in 1910. Whitehead and Rusell also used $\Lambda$ to denote the null class on page 229 in 1910. The following words improved from comments of @Goldstern may show why they made such uses.

$\mathrm{V}=\{x\mid x=x\}$, i.e., $\mathrm{V}=\{x\mid$ $x=x$ is true (verum)$\}$.

After them, people began to use $\mathrm{V}$ to denote the class of all individuals (objects). And nowadays there may be two more interpretations. While I don't know who gave those first.
(3) $\mathrm{V}$ is the v in von Neumann. This is because the cumulative universe is also called von Neumann universe which was first claimed by Gregory H. Moore in 1982 because existence and uniqueness of the general transfinite recursive definition of sets was demonstrated by von Neumann for both Zermelo–Fraenkel set theory in 1928 and his own set theory on pages 745–752 in 1928a. While in fact the first publication of the von Neumann universe was by Ernst Zermelo in 1930. Furthermore, the $\mathrm{V}$ notation was not used by von Neumann in his 1920s papers.
(4) $\mathrm{V}$ is the shape of the picture of the cumulative universe.
A: I heard from Kai Hauser that the letter $L$ comes from "law", and it is because the model is constructed using some laws.
