Ken Brown at Cornell Ordinal Numbers

http://www.math.cornell.edu/~kbrown/6310/ordinals.pdf

The search

“set theory” “ordinal numbers”

brings up the above web page in page 2 of Google results going by 10 at a time.

This does the list version of ghost sets.

The following is often called the von Neumann Construction.

So

0 = null

1 = {0} = {null}

2 = {0,1} = {0,{0}}

3 = {0,1,2} = {0,{0},{0,{1}}}

So we see a successor chain by set inclusion.

We have suggested ghost chain for any chain of sets built up from the null set by such means.

This growing list chain or set growth chain is not the same as the vertical or pure bracket Zermelo Construction or bracket construction or ghost bracket chain.  There we just add a pair of brackets as we increment.

The list approach is easier to understand in some ways.  It seems less fake as well.  This may be one of the motives that led von Neumann to it.

In the list approach, we have inclusion by order in a way that is closer to naive ways of thinking.

In the pure bracket VNC approach, we have inclusion by layers of brackets which is actually not something we deal with regularly and so it is an alien thought.