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I don't know how to state the question more clear: why it is so hard to compute immediate previous ordinal in von Neumann representation?

This is about Better representaion of natural numbers as sets?

To compute "next" number for $A$ we should just construct $A \cup \{A\}$ while to compute "previous" we should enumerate all members and find largest.

May be there is a better algorithm?

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1 Answer 1

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Note that if $\alpha$ is a von Neumann ordinal then $(\alpha,\subseteq)$ is a linear ordering. If it has a maximal element then it is the predecessor of $\alpha$, otherwise either $\alpha=0$ or it is a limit ordinal.

The maximum element of a chain of subsets is taken as the union of the chain, so let $\beta=\bigcup\alpha$, if $\beta\neq\alpha$ then $\alpha=\beta+1$.

I want to add that we hardly ever do this by hand. If $\alpha$ is a non-zero and non-limit then it must has a predecessor, call it $\beta$ and we are done. There is no actual computing involved.

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