In fact, we can additionally require that they appear as new labels in terms with isomorphic new parts. Even the barest form of this conclusion is enough to require large cardinals. The semi-finite Propositions 2.3 and 2.4 have obvious equivalent finite forms by compactness. Note that all of the predecessors of a vertex appear when that vertex is new. We now wish to consider construction sequences which are minimal in an appropriate sense.
Analogous results hold starting from a given poset.
We now present an independence result about infinite construction sequences. In fact, we can require that they appear as new vertices in terms with isomorphic new parts. Here isomorphisms between posets are order preserving bijections between the underlying integers which preserve the relevant structure. In fact, we can require that they appear as new vertices in terms with isomorphic new parts. This should be equivalent to the 1-consistency of type theory, or Zermelo set theory with bounded separation.