  In classical set theory, there are many equivalent ways to introduce
ordinals. In a constructive setting, however, the different notions split
apart, with different advantages and disadvantages for each. We consider three
different notions of ordinals in homotopy type theory, and show how they relate
to each other: A notation system based on Cantor normal forms, a refined notion
of Brouwer trees (inductively generated by zero, successor and countable
limits), and wellfounded extensional orders. For Cantor normal forms, most
properties are decidable, whereas for wellfounded extensional transitive
orders, most are undecidable. Formulations for Brouwer trees are usually
partially decidable. We demonstrate that all three notions have properties
expected of ordinals: their order relations, although defined differently in
each case, are all extensional and wellfounded, and the usual ordinal
arithmetic operations can be defined in each case. We connect these notions by
constructing structure preserving embeddings of Cantor normal forms into
Brouwer trees, and of these in turn into wellfounded extensional orders. We
have formalised most of our results in cubical Agda.
