Showing there is only one isomorphism between well ordered sets using transfinite induction

Question

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do this I want to show by induction that there is only one way to define the mapping and that is: $$\forall\, x\in A: \varphi\left(x\right)=\min\left(B\backslash\varphi\left(\left\{ a\in A\,:\, a<_{1}x\right\} \right)\right)$$ Showing the base of the induction is no problem and the induction hypothesis would be that $$\forall\, y<_{1}x\,:\,\varphi\left(y\right)=\min\left(B\backslash\varphi\left(\left\{ a\in A\,:\, a<_{1}y\right\} \right)\right)$$ My problem is that I can't seem to manage to use the hypothesis in order to prove the step. I've found a couple of ways to show this definition of the mapping is indeed necessary but none of those used the induction hypothesis. My plan was to mark $$c_{0}=\min\left(B\backslash\varphi\left(\left\{ a\in A\,:\, a<_{1}x\right\} \right)\right)$$ Using the fact $\varphi$ is surjective take $y\in A$ such that $\varphi\left(y\right)=c_{0}$ and show that it can't be that $y<_{1}x$ or $x<_{1}y$ and thus necessarily $y=x$. If someone could give me a way to show either one of these are false using the induction hypothesis it would solve my problem.

Thanks a lot!

Answer 1

What you want to show is that if $f\colon A\to B$ is an isomorphism of the given well-orders then $f=\varphi$.

Suppose that for $x\in A$ we have that $f\upharpoonright\{a\in A\mid a<_1 x\}=\varphi\upharpoonright\{a\in A\mid a<_1 x\}$. If $f(x)=\varphi(x)$ then we may continue the induction; otherwise $\varphi(x)<f(x)$ (by the definition of $\varphi$). Since $f$ is a bijection between the sets there is some $x'$ such that $\varphi(x)=f(x')$. It is not possible that $x'<x$ because for $a<x$ we have $\varphi(a)=f(a)$; so $x<x'$ but then we have $f(x')<f(x)$ which is a contradiction to the fact that $f$ is an order isomorphism.

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From the assumption that the orders are isomorphic we deduce that there is some isomorphism and by the proof above it has to be $\varphi$.

Answer 2

You might find it easier simply to show that if $f,g:A\to B$ are order-isomorphisms, then $f=g$, using induction to show that $f(a)=g(a)$ for all $a\in A$. Clearly $f(\min A)=g(\min A)$. Suppose that $a\in A$ is $<_1$-minimal such that $f(a)\ne g(a)$. Without loss of generality assume that $f(a)<_2 g(a)$. Show that $f(a)\notin\operatorname{ran}g$ to get a contradiction.