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 $2.1$