User blog comment:Emlightened/Ordinals in Type Theory/@comment-11227630-20180128144447/@comment-27513631-20180129195239

The presence of Nat and Ord, is the difference, of course. And the closed terms of λ→ and λP (each with atoms) are identical, so their diagonalising function is the same.

And I should also have specified that the 'extension by dependent types' includes dependent sum (otherwise you can't get \(\Sigma_{n:\texttt{Nat}}\texttt{Ord}^n(\texttt{Nat})\), only messy double-negated analogues I'm not confident with working with), so that's going to be edited in now.