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If $\alpha$ is a countable limit ordinal, then we know that $\alpha$ can be represented as a limit of an increasing sequence, i.e. $$\alpha = \bigcup_{i \in \omega} \alpha_i$$ where $\alpha_0 < \alpha_1 < \alpha_2 < \cdots$ (see e.g. here).

I am interested in whether the sequence $\alpha_i$ can be defined in a canonical way given $\alpha$. Unfortunately, this is apparently impossible in general (see the comments to this unanswered question; see also this answer). Therefore my question is instead:

Question: What is the largest subset of countable ordinals, i.e. subset of $\omega_1$, for which this map can be defined canonically?

In particular, and to be more precise, the mapping I am looking for is the following one for small $\alpha$:

  • $\omega \mapsto 0, 1, 2, 3, \ldots$

  • $2\omega \mapsto \omega, \omega + 1, \omega + 2, \omega + 3, \ldots$

  • $\omega^2 \mapsto 0, \omega, 2 \omega, 3 \omega, \ldots$

  • $\omega^\omega \mapsto 1, \omega, \omega^2, \omega^3, \ldots$

  • $\epsilon_0 \mapsto \omega, \omega^\omega, \omega^{\omega^\omega}, \ldots$

  • $\epsilon_\omega \mapsto \epsilon_0, \epsilon_1, \epsilon_2, \ldots$

And extended to other intermediate ordinals in the straightforward way, e.g.

  • $\epsilon_1 + \omega^2 \mapsto \epsilon_1, \epsilon_1 + \omega, \epsilon_1 + 2 \omega, \epsilon_1 + 3 \omega, \ldots$

Perhaps something like $\omega_1^{CK}$ would work mentioned in the above answer. Or maybe we need something much smaller.

I realize that "canonical" is not entirely well-defined. I am simply asking if the map can be defined in a constructive way that is both unique and consistent with the examples above (for example, not using the axiom of choice), and perhaps which has other nice properties.

Other potentially related questions: 1, 2, 3.

Caleb Stanford
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  • I'm guessing you have already looked at fundamental sequences? – Alvaro Pintado May 01 '25 at 00:46
  • @AlvaroPintado Thanks, fundamental sequences looks very relevant! This is basically what I'm asking for. Wikipedia implies that there are multiple systems of fundamental sequences with different properties? What I'm after is then a large system of fundamental sequences that can be defined constructively and in a natural way for some large ordinal. This looks an excellent starting point to look for literature on this topic I think... – Caleb Stanford May 01 '25 at 01:42
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    You do have to make some choices, but I don't know enough to tell you the different properties they end up having (maybe in a few months). I think you'll find that a lot of ordinal notation systems are computable and have computable fundamental sequences (this is important for ordinal analysis). Feel free to contact me over email to discuss more. – Alvaro Pintado May 02 '25 at 11:09

2 Answers2

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Given a theory $T$, say that a countable ordinal $\alpha$ is $T$-uncountable iff there is a transitive $\mathcal{M}\models T$ such that $\mathcal{M}\models$ "$\alpha$ is uncountable." For reasonable $T$, I would argue that we can think of the first $T$-uncountable ordinal as being the first ordinal for which the "computational power" related to $T$ (whatever that is) cannot build fundamental sequences. One natural choice of $T$ here is $\mathsf{KP}$ (roughly corresponding to $\Sigma_1$-transfinite recursion); so I would tentatively propose the smallest $\mathsf{KP}$-uncountable ordinal as a candidate.

The rest of this answer draws on recent work of mine and a result of Farmer Schlutzenberg:

Another, possibly large, candidate comes from computable structure theory. Having canonical fundamental sequences for all ordinals up to $\alpha$ is, computationally, equivalent to having a canonical presentation of $\alpha$ - that is, a canonical relation $R$ on the natural numbers with ordertype $\alpha$. In both cases of course the word "canonical" is a bit vague; what I really mean is that there is a very nice way to convert from one to the other. Now just as $\Sigma_1$-transfinite recursion is a "reasonable" amount of computational power to play with when building well-orderings, so too (I would argue) is the functional $\mathcal{O}^X$ sending $X$ to the canonical $\Pi^1_1$-complete-in-$X$ set. The following three ordinals are known to be in nondecreasing order:

  • The smallest $\mathsf{KP}$-uncountable ordinal.

  • The smallest ordinal $\beta$ such that there are presentations $A,B$ of $\beta$ with no single presentation of $\beta$ computable in each of $\mathcal{O}^A$ and $\mathcal{O}^B$.

  • The smallest ordinal $\gamma$ such that there are $A,B$ of $\beta$ with no isomorphism between $A$ and $B$ computable from $\mathcal{O}^A\oplus\mathcal{O}^B$.

And this $\gamma$ is known to (exist and) be no larger than the smallest ordinal $\delta$ such that $L_{\delta^\boxplus}\models$ "There is no $\Sigma_3$ injection from $\delta$ to $\omega$," where $\delta^\boxplus$ is the next admissible above $\delta$.

Noah Schweber
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    Fascinating, thank you for the answer and references! Taking your example of $T = \mathsf{KP}$, and say we let the smallest $T$-uncountable ordinal be $\omega_{\mathsf{KP}}$, what would then be the corresponding map $f: \omega_{\mathsf{KP}} \to (\omega_{\mathsf{KP}})^\omega$, $\alpha \mapsto \alpha_0, \alpha_1, \alpha_2, \ldots$? – Caleb Stanford Aug 31 '24 at 03:04
  • @CalebStanford Fix a limit ordinal $\alpha<\omega_{\mathsf{KP}}$. By definition of $\omega_\mathsf{KP}$, the next admissible set $L_{\alpha^\boxplus}$ sees that $\alpha$ is countable. The canonical fundamental sequence for $\alpha$ is then the "$L$-least" (that is, first in the $L$-ordering) fundamental sequence for $\alpha$ living in $L_{\alpha^\boxplus}$. This idea works for all $T$ which are compatible with $\mathsf{V=L}$, or at least compatible with some formula defining a well-ordering of the universe. – Noah Schweber Aug 31 '24 at 19:56
  • Thanks! For reference: admissible set. I may need to review computable structure theory (Ash/Knight) for some of the rest :) For now can I take your answer as saying that the map can be defined for a large space of countable ordinals? – Caleb Stanford Sep 02 '24 at 13:40
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Unfortunately I believe your desire for a unique mapping might break down as soon as $\varepsilon_1$, which has two equally valid increasing sequences that I am aware of:

  • $\varepsilon_0+1, \omega^{\varepsilon_0+1}, \omega^{\omega^{\varepsilon_0+1}}, \ldots$

  • $\varepsilon_0, \varepsilon_0^{\varepsilon_0}, \varepsilon_0^{\varepsilon_0^{\varepsilon_0}}, \ldots$

Shophaune
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