Bracket chain of sets and numbers

Suppose 0 is something.  The Null. Whatever that is.

The number one is defined as the set that contains 0.

The number n’ is defined as the set that contains n as its

sole member.

So n’ = {n}

and 0 is some base object.

This is the Zermelo 1908 Construction.  This is according to Enderton Set Theory page 67.

Whatever the base object is, we get a bracket chain of sets. We can call this a bracket number chain.  It is a reference chain.

This defines each element of the chain.  We still have to define the initial segments or head segments.

So

H_n = {0,1,…,n}

Here H_n is not the same as ‘n, n or n’.

..

In the von Neumann Construction, we get head segments as part of the construction.  The VNC is the list of prior sets (which he calls numbers).

 

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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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