Question on countable ordinals.

Question

Is it possible to select for each countable (non-zero) limit ordinal $\alpha$ an ordinal $x_\alpha<\alpha$ in such a way that the $x_\alpha$ are distinct? I have no idea how to go about this. If is true, then what happens if we allow uncountable ordinals?

Answer 1

EDIT: I missed that the question is only about limit ordinals, but Fodor still applies since the set of limit ordinals is stationary, and my argument below the fold also still goes through (with a slight tweak - we need $\delta_n$ to be sufficiently large and in the domain of $f$) since the set of limit ordinals is a club.

Well, you get into a problem with $\alpha=0$. :P Similarly, if you allow finite nonzero ordinals, then you get into a problem at $\omega$: $1$ has to go to $0$, then $2$ has to go to $1$, then $3$ has to go to $2$, ... , and that leaves nothing for $\omega$ to go to.

So the right way to ask the question is to restrict attention to "sufficiently large" countable ordinals - say, infinite ones. This seems to get around the immediate problem ...

... but now we run into a more subtle one, essentially an elaboration on that described above: Fodor's lemma. The function $\alpha\mapsto x_\alpha$ is regressive, and even after restricting to "sufficiently large" countable ordinals is still defined on a stationary subset of $\omega_1$ (since every cobounded set is stationary) and so is constant on a "large" set.

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Actually, bringing Fodor in here is a bit overkill; the relevant case of Fodor's lemma here is easier to prove than the whole. Suppose we have a regressive 1-1 function $f: \alpha\mapsto x_\alpha$ defined for all $\alpha\in [\lambda,\omega_1)$ for some countable $\lambda$.

Let's think about when various ordinals show up in the image of $f$: for $\beta<\omega_1$, let $Time(\beta)$ be the least ordinal $\gamma$ such that for all $\beta'\le \beta$, if $\beta'\in im(f)$ then $f^{-1}(\beta')<\gamma$.

Now define the sequence

• $\delta_0=\lambda$,

• $\delta_{n+1}=Time(\delta_n)$.

Let $\delta=\lim_{n\rightarrow\infty}\delta_n$; now, what can $f(\delta)$ be?