Gödel codes with rounded corners?

Question

The simplest way to produce Gödel codes is with $\ulcorner$ and $\urcorner$, and we may write $\ulcorner p\urcorner$ to enclose the formula $p$ within codes.

Buss programmed a macro which gives a better appearance than Gödel codes in many contextst.

The macro Godeln{}, which egreg recently suggested, as an aside in an answer to a recent question (Dotted x in mathmode), is an improvement of Buss's improvements of \(\ulcorner \urcorner\).

My question is: May the corners of the egreg codes be rounded, e.g. with TikZ, to obtain codes which are discernibly different?

MWE

\documentclass[]{article}
\usepackage{amsmath, amssymb}
\newlength{\gnCornerHgt}
\newlength{\gnArgHgt}
\makeatletter
\newcommand{\Godeln}[1]{\mathinner{\mathpalette\Godeln@{#1}}}
\newcommand{\Godeln@}[2]{%
    \begingroup
    \settoheight{\gnCornerHgt}{$\m@th#1\ulcorner$}%
    \settoheight{\gnArgHgt}{$\m@th#1#2$}%
    \ifdim\gnArgHgt<\gnCornerHgt
    \setlength{\gnArgHgt}{0pt}%
    \else
    \addtolength{\gnArgHgt}{-\gnCornerHgt}%
    \fi 
    \raisebox{\gnArgHgt}{$\m@th#1\ulcorner$}%
    #2%
    \raisebox\gnArgHgt{$\m@th#1\urcorner$}%
    \endgroup
}
\makeatother
\begin{document}
$\ulcorner 2^2\urcorner$ vs. $\Godeln{2^2}$



\end{document}

Answer 1

Here is a somewhat simpler draw-it-yourself version with TikZ. You can adjust the radius of the rounded corners by changing the .5mm in the \draw commands. You can adjust the horizontal spacing by changing inner xsep, and the vertical spacing by changing inner ysep.

There is also an optional argument that can be used to change the \draw style. For example, \godel[thick]{a}. Colors work too.

\documentclass{article}
\usepackage{tikz}
\DeclareRobustCommand{\godel}[2][]{{}\mathbin{\vcenter{\hbox{\tikz{
  \nodeinner ysep=-1pt, inner xsep=3pt{#2\strut};
  \draw[rounded corners=.5mm, #1]([xshift=-.3em]M.north east)--(M.north east)--++(0,-.3em);
  \draw[rounded corners=.5mm, #1]([xshift=.3em]M.north west)--(M.north west)--++(0,-.3em);
}}}}{}}
\begin{document}
Some examples: $\godel{a}\godel{A}\godel{g}\godel{(}$
\section{Here is a G"odel code $\godel[thick]{a}$}
\end{document}

Answer 2

A variant of Sandy G's code. The main differences are

1. how the baseline is treated, so we're guaranteed that the symbol has the correct one

2. the argument to \godel is typeset in math mode.

3. the sidebearings are fixed and don't shrink nor stretch

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\newcommand{\sandygodel}[2][]{{}\mathbin{\vcenter{\hbox{\tikz{
  \nodeinner ysep=-1pt, inner xsep=3pt{#2\strut};
  \draw[rounded corners=.5mm, #1]([xshift=-.3em]M.north east)--(M.north east)--++(0,-.3em);
  \draw[rounded corners=.5mm, #1]([xshift=.3em]M.north west)--(M.north west)--++(0,-.3em);
}}}}{}}
\newcommand{\godel}[2][]{%
  \mspace{1\medmuskip}%
  \vphantom{#2}%
  \begin{tikzpicture}[baseline=(M.south)]
    \nodeinner ysep=0pt,inner xsep=3pt{\smash[b]{$#2\mathstrut$}};
    \draw[rounded corners=.5mm,#1]([xshift=-1mm]M.north east)--(M.north east)--++(0,-.1);
    \draw[rounded corners=.5mm,#1]([xshift=1mm]M.north west)--(M.north west)--++(0,-.1);
  \end{tikzpicture}%
  \mspace{1\medmuskip}%
}
\begin{document}
\tableofcontents
\section{Egreg's $\godel{A}$ and $\godel[thick]{A}$}
Some examples: $\godel{a}\godel{A}\godel{g}\godel{(}$
Squeezed: \hbox spread -10pt{$\godel{a}\godel{A}$}
Math: $\godel{r\in R}$
\medskip
{\Large Some examples: $\godel{a}\godel{A}\godel{g}\godel{(}$}
\section{Sandy's $\sandygodel{A}$ and $\sandygodel[thick]{A}$}
Some examples: $\sandygodel{a}\sandygodel{A}\sandygodel{g}\sandygodel{(}$
Squeezed: \hbox spread -10pt{$\sandygodel{a}\sandygodel{A}$}
Math: $\sandygodel{$r\in R$}$
\medskip
{\Large Some examples: $\sandygodel{a}\sandygodel{A}\sandygodel{g}\sandygodel{(}$}
\end{document}