From a constructive and computational perspective, what are the ordinal numbers?

On the one hand, it seems you can represent ordinal numbers symbolically using something like Cantor Normal Form notation. Then the ordinals are discrete, symbolic entities, and things like "$\leq$" become decidable.

On the other hand, you can represent the ordinals recursively as monotonically nondecreasing sequences of smaller ordinals, and then define an undecidable $\leq$ relation: An ordinal $(\alpha_i)$ is greater than or equal to $(\beta_i)$ if for every element of $\beta_j$ there is an element $\alpha_i$ such that $\alpha_i \geq \beta_j$. I struggle to see how this could be useful though.

nLab also mentions the Plump Ordinals, but I'm not sure what they are.

For an application, let $\mathrm{On}$, denote the class of all ordinals. Let $\alpha : \mathrm{On} \to \mathbb{R}$ be a mapping from $\mathrm{On}$ to real numbers. For any finite set $S = \{i_1, \ldots, i_n\} \subseteq \mathrm{On}$ where $i_1 < \cdots < i_n$, define the quantity
$${\mathcal K}_S(\alpha) = \sqrt{\alpha_{i_1}^{2^1} +
\sqrt{\alpha_{i_2}^{2^2} + \cdots \sqrt{\alpha_{i_n}^{2^n}}}
}$$
Say that a real $\mathcal{K}(\alpha)$ is the *limit* of $S \mapsto \mathcal{K}_S(\alpha)$ when for every $\epsilon > 0$ there exists a finite $S \subseteq \mathrm{On}$ such that for all finite $T \subseteq \mathrm{On}$, if $S \subseteq T$ then $|\mathcal{K}_S(\alpha) - \mathcal{K}(\alpha)| < \epsilon$.

Examples:

if $\alpha_n = 2$ then $\mathcal{K}(\alpha) = \sqrt{2^{2^1} + \sqrt{2^{2^2} + \sqrt{2^{2^3} + \dotsb}}} = 2\phi$ where $\phi$ is the Golden ratio.

given $x \in \mathbb{R}$, take $$\alpha_n = \begin{cases} 1 & \text{if $n \neq \omega$,}\\ x & \text{if $n = \omega$} \end{cases}$$ Then $\mathcal{K}(\alpha)$ is the limit if the sequence $x, \sqrt{1+x^2}, \sqrt{1+\sqrt{1+x^4}}, \sqrt{1+\sqrt{1+\sqrt{1+x^8}}}, \ldots$. This is a continued radical which is "transfinite".

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