Ghost chains and von Neumann construction

If you start with the empty set or null set, and build up sets from that, you can all them ghost sets and the chains of sets, ghost chains.

So the VNC is a ghost chain of ghost sets.

The following construction is the Zermelo construction according to Enderton Set Theory page 67.

This one is made using brackets only after the base is used, so it can be called a bracket ghost chain or bracket chain as well.

null

{null}

{{null}}
each successor in the bracket ghost chain is the set consisting of the prior set as its single member.

Single member can be defined in logic.

A set has a single member if exists x such that if y is in said set, then y equals x.

We can also consider the list approach starting with the null set.

An example element is

{null, {null},{null,{null}}}

This set is a more typical type of inclusion set.

The bracket set is a different type of inclusion chain.

We can think of the pure bracket, vertical VNC, approach as a ladder going up.

The list approach might be horizontal or it might be diagonal in a sense.  The list approach is called the VNC in many webpages and references.  The pure bracket approach also is given that name in at least one reference, I believe.  However, it should be called the Zermelo construction according to Enderton Set Theory page 67.

A pure horizontal might be using little balls with number labels already on them.  This is the conventional way of thinking.

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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