Constructive bijection from fundamental sequence

Question

In this article I found the following statement: "The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence which has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α, (i.e. one not using the axiom of choice)". But how exactly this constructive bijection is constructed?

Answer 1

What the author of that sentence had in mind must be something like:

Without loss of generality, your assumed fundamental sequence $(\beta_i)_{i\in\omega}$ for $\alpha$ starts with $\beta_0=0$ and otherwise contains only limit ordinals. (If this is not the case, then replace every non-limit ordinal with the largest limit less than it, and remove duplicates from the sequence). (Some special pleading will be necessary if $\alpha$ has the form $\beta+\omega$ for some $\beta$, but that is a boring case too).

The elemements of the sequence now breaks $\alpha$ into $\omega$ many intervals $[\beta_i,\beta_{i+1})$. Each of these intervals is well-ordered with an order type less than $\alpha$. Since you have chosen fundamental sequences for all smaller limit ordinals, you can recursively define bijections between each $[\beta_i,\beta_{i+1})$ and $\omega$. Now use your favorite bijection $\omega\times\omega\to\omega$ to combine all of these bijections into a single bijection $\alpha\to\omega$.

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What is this good for? I'm not entirely sure. The purpose of the statement in the article seems to be mostly motivational: using this construction we can use the Veblen functions to construct an explicit enumeration of all ordinals up to some large countable ordinal, and that is one reason to care about those functions.

The claim almost certainly doesn't mean that there is a single definable concept of "fundamental sequence" that pinpoints exactly one sequence with each $\omega$-cofinal ordinal as its limit (or even just for each countable limit ordinal).