Back-n-Forth or a Direct Isomorphism Between A Countable DLO Without Endpoints and $U = \{ \frac{m}{2^n} \}$

Question

Note: I changed this question and deleted my answer, bringing it into the question for quick review. The proof now has the same brevity as the back-and-forth method found in wikipedia.

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Let $P$ be a Countable DLO Without Endpoints.

Let $U = \{m2^{-n} \; | \; \text{ with } m \in \mathbb Z \text{ and } n \in \mathbb N\}$

Here is proof showing that there is a order-preserving bijection $\tau$ of $P$ onto $U$.

Let $P$ be bijectively enumerated by $(p_n)_{n \ge 0}$. We will define the mapping $\tau$ by 'running' through this sequence one term at a time.

We map $p_0 \mapsto 0$. Suppose we've injectively mapped (order preserving) the terms $(p_j)_{j \lt k}$ into $U$. If $p_k$ is 'outside' the interval range of $(p_j)_{j \lt k}$ in $P$, then we select the extreme $p_{j_0}$ and map $p_k$ to either $\tau(p_{j_0}) + 1$ or $\tau(p_{j_0}) - 1$ thereby keeping $\tau$ order-preserving. If $p_k$ is 'inside' the current $\tau$ range, it has a predecessor $p_{j_0}$ and a successor $p_{j_1}$, so that we can satisfactorily map $p_k$ to

$\quad \frac{\tau(p_{j_0}) + \tau(p_{j_1})}{2}$

Using the hypotheses on $P$ and properties of $U$, we can easily see that the mapping $\tau$ extends to a surjection making $\tau$ an isomorphism.

As a disclaimer I know nothing about model theory (see first comment below this question), but

Can a different proof than the back-and-forth method be of use in the field of model theory?

Is their a simple/universal/canonical example that one learns about in model theory?

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Note: The proof technique gives corresponding constructs when $P$ has a maximum or minimum - all the stuff in $U$ bounded by any maximum or minimum still gets filled in.

Answer 1

Let $(p_n\mid n\in\Bbb N)$ be a enumeration of $P$.

Lemma: Let $(P,<)$ be countable, dense, and linearly ordered set without endpoints. Then there is a order embedding $f:\Bbb Z \to P$ such that $f[\Bbb Z]$ is unbounded from above and from below in $P$.

Proof:

We define a mapping $g:\Bbb N \to \Bbb N$ recursively by $$g(0)=0 \text{ and }g(n+1)=\min \{i\in\Bbb N\mid p_{g(n)}<p_i\}$$

We define a mapping $f_1:\Bbb N \to P$ by $$f_1(n)=p_{g(n)}$$

It follows from the definition of $f_1$ that $\forall n\in\Bbb N:p_{g(n)}<p_{g(n+1)}$ and thus $f_1$ is injective. Let $A:=f_1[\Bbb N]$.

$p_0=f_1(0)<f_1(1) \implies A$ is unbounded above by $p_0$. Assume that $A$ is unbounded above by $p_i$ for all $i\le n$. Then $\exists n_0\in \Bbb N,\forall i\le n:p_i \le f_1(n_0)= p_{g(n_0)}$.

• If $p_{n+1}\le p_{g(n_0)}=f_1(n_0)$: $A$ is unbounded above by $p_{n+1}$.

• If $p_{n+1} > p_{g(n_0)}$: We have $\forall i\le n:p_i\le p_{g(n_0)}$ and $p_{g(n_0)}<p_{n+1} \implies$ $\min \{i\in\Bbb N\mid p_{g(n_0)}<p_i\}=n+1 \implies g(n_0+1)=n+1 \implies$ $p_{g(n_0+1)} = p_{n+1} \implies f_1(n_0+1)=p_{n+1}$. Thus $A$ is unbounded above by $p_{n+1}$.

Hence $f_1[\Bbb N]$ is unbounded from above in $P$.

We define a reverse-order $<^*$ on $P$ by $\forall x,y\in P:x <^* y \iff y<x$. Then $(P,<^*)$ is a countable, dense, and linearly ordered set without endpoints.

In a similar manner, we obtain $f_2:\Bbb N \to P$ which is an order embedding from $\Bbb N$ to $P$ such that $f_2(0)=p_0$ and that $f_2[\Bbb N]$ is unbounded from above in $P$ with respect to $<^*$. Thus $f_2[\Bbb N]$ is unbounded from below in $P$ with respect to $<$.

Let $f=f_1\cup f_2$. It is easy to verify that $f:\Bbb Z \to P$ is order embedding from $\Bbb Z$ to $P$ such that $f[\Bbb Z]$ is unbounded from above and from below in $P$.$\quad \blacksquare$

Theorem: There exists an order isomorphism between $U = \left\{\dfrac{m}{2^n} \mid m \in \mathbb Z \text{ and } n \in \mathbb N\right\}$ and $P$.

Proof:

Let $U_k = \left\{\dfrac{m}{2^k} \mid m \in \mathbb Z\right\}$ for all $k\in\Bbb N$. It's clear that $U_0=\Bbb Z$, that $U_k\subsetneq U_{k+1}$ for all $k\in\Bbb N$, and that $U_k$ is unbounded from above and from below in $\Bbb Q$.

We define recursively a family of mappings $(F_k\mid k\in\Bbb N)$ such that $F_k$ is an order embedding from $U_k$ to $P$, and that $F_k\subsetneq F_{k+1}$ for all $k\in\Bbb N$.

Let $F_0=f$ where $f$ is generated by Lemma.

Assume that we have defined $F_k$, we define $F_{k+1}$ as follows:

• $F_{k+1}\restriction U_k:=F_k$.

• For each $z\in U_{k+1}\setminus U_k$, there is a unique $m\in\Bbb Z$ such that $\dfrac{m}{2^k}<z<\dfrac{m+1}{2^k}$ since $U_k$ is unbounded from above and from below in $\Bbb Q$. Let $F_{k+1}(z):=p_{i_0}$ where $i_0=\min \{i\in\Bbb N\mid F_k(\frac{m}{2^{k}})<p_i<F_k(\frac{m+1}{2^{k}})\}$. Since $P$ is dense, such $i_0$ does exists. Thus $F_{k+1}(z)$ is well-defined for all $z\in U_{k+1}\setminus U_k$.

Let $F=\bigcup_{k\in\Bbb N}F_k$. It is easy to verify that $F$ is an order embedding from $\bigcup_{k\in\Bbb N}U_k=U$ to $P$.

Let $\bigcup_{k\in\Bbb N}F_k[U_k]=P'$. We next prove that $\forall n\in\Bbb N:p_n\in P'$ by strong induction on $n$.

It's clear that $p_0\in f[\Bbb Z]=F_0[U_0]$. Thus $p_0\in P'$. Assume that $p_i\in P'$ for all $i\le n$. Then there exists $k\in\Bbb N$ such that $p_i\in F_k[U_k]$ for all $i\le n$.

1. $p_{n+1} \in F_k[U_k]$

Then $p_{n+1}\in P'$.

2. $p_{n+1} \notin F_k[U_k]$

Then there is a unique $m\in\Bbb Z$ such that $F_k(\frac{m}{2^k})<p_{n+1}<F_k(\frac{m+1}{2^{k}})$ where$\frac{m}{2^k}\in U_k$ by the fact that $F_k[U_k]$ is unbounded from above and from below in $P$.

We have $\forall i\le n:p_i\in F_k[U_k] \implies \forall i\le n:i\notin \{i\in\Bbb N \mid F_k(\frac{m}{2^{k}})<p_i<F_k(\frac{m+1}{2^{k}})\}$ by the fact that $F_k$ is an order isomorphism between $U_k$ and $F_k[U_k]$, and that $\frac{m}{2^{k}}$ and $\frac{m+1}{2^{k}}$ are two consecutive members of $U_k$.

Moreover, $F_k(\frac{m}{2^k})<p_{n+1}<F_k(\frac{m+1}{2^{k}}) \implies n+1\in \{i\in\Bbb N\mid F_k(\frac{m}{2^{k}})<p_i<F_k(\frac{m+1}{2^{k}})\}$ $\implies n+1=\min \{i\in\Bbb N\mid F_k(\frac{m}{2^{k}})<p_i<F_k(\frac{m+1}{2^{k}})\} \implies F_{k+1}(z)=p_{n+1}$ where $z\in U_{k+1}\setminus U_k$ such that $\dfrac{m}{2^k}<z<\dfrac{m+1}{2^k} \implies p_{n+1}\in F_{k+1}[U_{k+1}] \implies p_{n+1}\in P'$.

By principle of strong induction, $P\subseteq P'$. Furthermore, $P'\subseteq P$. Thus $P=P'$. Hence $F$ is an order isomorphism between $U$ and $P$.$\quad \blacksquare$