But we believe it is better to follow the treatment in the current posting rather than 30. Thus in particular, G is an induced subgraph of G'. The valence of the new vertex x in G' is the number of vertices that are connected to x by an edge in G'. Note that this valence is one less than the same as the number of vertices in the new part, and is also the valence of the new vertex in the new part. We now wish to consider construction sequences which are minimal in an appropriate sense.
Analogous results hold starting from a given graph.
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 graphs 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.