# Order Automorphism on Well-Ordered Class is Forward Moving

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## Theorem

Let $\struct {A, \preccurlyeq}$ be a well-ordered class.

Let $\phi$ be an order isomorphism on $\struct {A, \preccurlyeq}$.

Then:

- $\forall a \in A: a \preccurlyeq \map \phi a$

## Proof

Let us define an element $a$ of $A$ such that:

- $\map \phi a \prec a$

as **moving backwards**.

Aiming for a contradiction, suppose there exists an element $a$ of $A$ that **moves backwards**:

- $\map \phi a \prec a$

for some $a \in A$.

Then applying $\phi$ to both sides:

- $\map \phi {\map \phi a} \prec \map \phi a$

That is:

- $\map \phi a$ also
**moves backwards**

Thus if some $a \in A$ **moves backwards**, there is another predecessor element that also **moves backwards**.

Hence there is no smallest element of the set of all elements of $A$ that **move backwards**.

But this contradicts the properties of a well-ordered class:

- every non-empty subclass of $A$ has a smallest element under $\preccurlyeq$.

Hence there can be no elements of $A$ that **move backwards**.

That is:

- $\forall a \in A: a \preccurlyeq \map \phi a$

$\blacksquare$

## Linguistic Note

The term **forward moving** was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

The concept is introduced by Raymond M. Smullyan and Melvin Fitting briefly in their *Set Theory and the Continuum Problem, revised ed.* of $2010$ as a stepping-stone to the stronger result Order Automorphism on Well-Ordered Class is Identity Mapping.

They do not actually give a name to the concept, but merely characterise it as a class mapping under which no element **moves backwards**.

It is worth comparing with the concept of a **progressing mapping**.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings: Theorem $2.1$