\documentclass[12pt]{article}
\usepackage{mymacros}

%1.5: 1,6
%1.6: 3,7
%1.7: 1,5

\begin{document}
\title{1.5}
\author{Andrew Lounsbury}
\date{February 4, 2020}
\maketitle

\section{1.5, p.21}

\begin{enumerate}
	\item[1.]
	\begin{enumerate}
	\item
	\underline{$(\forall x)(\forall y)(x+y = 2)$}
	
	free variables: none, so this is a sentence
	
	\item
	\underline{$(x+y < x) \lor (\forall z)(z < 0)$}
	
	free variables: $x,y$, so this is not a sentence
	
	\item
	\underline{$((\forall y)(y < x)) \lor ((\forall x)(x < y))$}
	
	free variables: $x,y$, so this is not a sentence
	
	\end{enumerate}

	\item[6.]
	$\phi(x) :\equiv \underbrace{[(\forall y)(x = y)]}_{\alpha} \lor \underbrace{[(\forall x)(x < 0)]}_{\beta}$, $t:\equiv S0$

	The variable $x$ is only free in $\alpha$, so $\phi(t) :\equiv [(\forall y)(t = y)] \lor [(\forall x)(x < 0)]$

\end{enumerate}

\section{1.6, p.26}

\begin{enumerate}
	\item[3.]
	$\Ell$ is $\{\flat,\sharp^3,\natural^2\}$

	\item[7.]
	$\Ell_{NT}$ is $\{0,S,+,\cdot,E,<\}$ \\
	$\fN = (\N,0^{\fN},S^{\fN},+^{\fN},\cdot^{\fN},E^{\fN},<^{\fN})$ \\
	$S^{\fN}(t) :\equiv St$ \\
	$+^{\fN}(t,s) :\equiv +ts$ \\
	$\cdot^{\fN}(t,s) :\equiv \cdot ts$ \\
	$E^{\fN}(t,s) :\equiv Ets$ \\
	$<^{\fN}(t,s) :\equiv <ts$

	$S0+S0 \stackrel{?}= SS0$

\end{enumerate}

\section{1.7, p.32}

\begin{enumerate}
	\item[1.]
	The structure $\fN$ makes the sentence $1+1 = 2$ true. 
	\begin{proof}

	\end{proof}

	\item[5.]
	Let $\fA$ be a structure for the language of set theory, $\Ell_{ST}$, which is $\{\in\}$. Let $A = \{u,v,w,\{u\},\{u,v\},\{u,v,w\}\}$. 

	$(\fa y \in y)(\ex x \in x)(x = y)$

\end{enumerate}
\end{document}