$\mathbb{N}\times\{0,1\}$ and $\{0,1\}\times \mathbb{N}$ not isomorphic

Question

Show that the sets $W=\mathbb{N}\times\{0,1\}$ and $W'=\{0,1\}\times\mathbb{N}$, ordered lexicographically are non-isomorphic well-ordered sets.

Any hints on how to prove this?

Answer 1

UPD: changed ordering convention.

Consider an element $y=(1, 0) \in W'$. It has infinite number of predecessors. In $W$ every element has only finite number of predecessors.

Now assume there is an order preserving isomorphism $f : W \mapsto W'$. Take $x \in W$ such that $f(x)=y$. Let $X= \{x' \in W | x' < x\}$ and $Y= \{y' \in W' | y' < y\}$ and $X'= \{f^{-1}(y) \in W | y \in Y\}$. Since $f$ in an isomorphism there is bijection between $Y$ and $X'$. Since $f$ preserves order, $X' \subset X$. So $|Y| = |X'| \le |X|$, a contradiction since Y is not finite.

Answer 2

I'm posting this to practice MathJax, so ignore.

$$ \begin{matrix} \bullet & \rightarrow& \bullet\\ & \swarrow\\ \bullet & \rightarrow& \bullet\\ & \swarrow\\ \bullet & \rightarrow& \bullet\\ & \swarrow\\ \bullet & \rightarrow& \bullet\\ & \swarrow\\ \bullet & \rightarrow& \bullet\\ & \swarrow\\ &\cdots\\ &\cdots \end{matrix} $$

$$ \begin{matrix} \bullet & \rightarrow& \bullet & \rightarrow &\bullet & \rightarrow &\bullet & \rightarrow &\bullet & \rightarrow &\bullet \rightarrow \cdots\\ &&&&&&&&&&\swarrow\\ &&\longleftarrow&\longleftarrow&\longleftarrow&\longleftarrow&\longleftarrow&\longleftarrow&\longleftarrow&\longleftarrow\\ &\swarrow\\ \bullet & \rightarrow& \bullet & \rightarrow &\bullet & \rightarrow &\bullet & \rightarrow &\bullet & \rightarrow &\bullet \rightarrow \cdots\\ \end{matrix} $$