$3\mathrm{.}7\mathrm{.}12*$ Consider the logical system NL $("$Number Logic"$)$ that is obtained from PL by

making the following modifications to PL syntax and semantics: $(1)$ the names of NL

include infinitely many new names ${}_0,{}_1,{}_2,{}_3,$dots,${}_n,$dots; $(2)$ the vocabulary of NL contains

an additional $1-$place predicate tilde$(N)$; $(3)$ the list of names of every NL interpretation includes

the names ${}_0,{}_1,{}_2,{}_3,$dots,${}_n,$dots; $(4)$ the universe of discourse of every $NL$ interpretation

contains every natural number, that is$,$ NsubeUD; $(5)$ every $NL$ interpretation assigns the

natural number $n$ as the referent of the name ${}_n$; and $(6)$ every $NL$ interpretation assigns $N$

as the extension of the predicate tilde$(N).$ All the logical concepts in NL are defined exactly as in

PL$.$ Prove the following.

$3\mathrm{.}7\mathrm{.}12$a The sentences $(AAx)(AAy)x=y$ and $(EEx)(EEy)(x{y}^{{?}^{?}}(AAz)(z=xvvz=y))$ are

contradictory in NL and every sentence of the form tilde$(N){}_n$ is valid in $NL.$

$3\mathrm{.}7\mathrm{.}12$b If the proof theory of PL is introduced as a proof theory for $NL,$ then $NL$

proof theory is sound but incomplete.