A formula in the first$-$order language of arithmetic is defined induc$-$

tively as follows:

$$ if t$1$ and t$2$ are terms, then t$1=$ t$2$ is a formula,

$$ if t$1$ and t$2$ are terms, then t$1<$ t$2$ is a formula,

$$ if $\backslash$phi is a formula, then $\backslash$phi is a formula,

$$ if $\backslash$phi and $\backslash$psi are formulas, then $\backslash$phi $\backslash$psi is a formula, and

$$ if $\backslash$phi is a formula, then $$x$[$w$]\backslash$phi is a formula whenever w is a number

$($written in binary$).$

Show that $\{\backslash$phi : $\backslash$phi is a formula $\}$ is context$-$free.

SOLUTION: