Lower Sections of Well-Ordered Classes are Order Isomorphic at most Uniquely
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Theorem
Let $\struct {A, \preccurlyeq_A}$ and $\struct {B, \preccurlyeq_B}$ be well-ordered classes.
Let $L_A$ be a lower section of $\struct {A, \preccurlyeq_A}$.
Then:
- $(1): \quad L_A$ is order isomorphic to at most one lower section $L_B$ of $\struct {B, \preccurlyeq_B}$
- $(2): \quad$ If such an $L_B$ exists, there exists exactly one order isomorphism from $L_A$ to $L_B$.
Proof
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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: Corollary $2.7$