Prove that Principle $2$ implies that we have infinitely many

stages available.

Hint. Arguing by contradiction, assume instead that we only have finitely

many stages. So repeatedly applying Principle $2$ we can form a non end$-$

ing sequence of stages

${}^{\text{'}\text{'}}dot{s}^{\text{'}}{}^{\text{'}\text{'}}dot{s}^{\text{'}}$