$2\mathrm{.}5\mathrm{.}13$ Suppose that we have a logical system that allows us to form infinite conjunctions

of any sentences. For instance, in this logic the infinite conjunctions

$P{a}^{{?}^{?}}notP{a}^{{?}^{?}}P{a}^{{?}^{?}}notP{a}^{{?}^{?}}P{a}^{{?}^{?}}notP{a}^{{?}^{?}}$dots and $P{a}^{{?}^{?}}P{a}^{{?}^{?}}notP{a}^{{?}^{?}}P{a}^{{?}^{?}}P{a}^{{?}^{?}}notP{a}^{{?}^{?}}P{a}^{{?}^{?}}P{a}^{{?}^{?}}notP{a}^{{?}^{?}}$dots are well$-$

formed sentences. Use diagonalization to prove that the set whose members are all the

infinite conjunctions that are made of any combinations of $Pa$ and notPa is uncountable.