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\begin{document}
\title{5.10-6.2 Questions}
\author{Andrew Lounsbury}
\date{April 6, 2020}
\maketitle

\section{5.10, p.153}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
    \item[Thm 5.10.2]
    Correct me if I'm wrong. $BaseCaseSet$  is the set of strings (terms, formulas, expressions?) used in the first steps of a recursive definition (for instance, the atomic formulas) which are then used to define the recursive part of the definition (e.g., $\neg\alpha, \alpha \lor \beta$, and $(\fa x)\alpha$), and then \textsc{Basecaseset} is the set of G\"odel numbers of those strings. 
\end{enumerate}

\section{5.11, p.156}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
    \item[p.158, top of page]
    It's a bit confusing that they use $x$ for two different things here. It would make more sense to me to write E1 as $v = v$ for each variable $v$, and then suppose $\ulcorner v \urcorner = x$. Then, 
    \begin{align*}
        \ulcorner v = v \urcorner 
        &= \left<7, \ulcorner v \urcorner, \ulcorner v \urcorner\right> \\
        &= \left<7, x, x \right> \\
        &= 2^8 3^{x+1} 5^{x+1} \\
        &= 2^8 3^{Sx} 5^{Sx}. 
    \end{align*}
    The way they have it in the book initially made me think of the exponents of $3^{Sx}$ and $5^{Sx}$ as $S$ applied to a variable, but then these exponents wouldn't be natural numbers. 
\end{enumerate}

\section{5.12, p.158}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I didn't mark any questions for section 5.12. 

\section{6.2, p.171}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
    \item[p.171 (iii)]
    Could you reiterate the meaning of `` $\vdash \left[(\fa i < y)[\neg R(\ol{a},i)] \rightarrow \cdots\right.$'', where the $\vdash$ has nothing on the left? I'm thinking it means that that formula is provable in every $\Ell$-structure, in comparison with their definition of `` $\vDash \phi$''from Definition 1.9.2 on p.37, but it has been a while since I last saw this. 
\end{enumerate}

\section{Random, probably dumb question}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In reading about these mathematical languages and the analysis of them, I can't help but think about how they're subliminally used elsewhere. Surely if a physicist happens to say $1+1 = 2$ or $\frac{dv}{dt}$ in their work, they're merely using the same machinery of  $\Ell_{NT}$/$\fN$ or the language of Calculus behind the scenes. Or, when a chemist does molecular geometry, surely they must be using the same old geometry from some language of geometry $\Ell_{Geo}$, though they may not realize it. Would it be possibly (but perhaps not practical at all) to lump a bunch of languages together into some sort of composite language for subject such as Physics as
\[
    \Ell_{Ph} \text{ is } \Ell_{NT} \cup \Ell_{Calc} \cup \Ell_{Geo} \cup \text{ (a bunch of other stuff physics uses)}? 
\]

\end{document}