Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapst …

Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how wid …

Dmytro Taranovsky has a Web page on which he claims to define a system of ordinal notations strong enough to provide an ordinal analysis of full second-order arithmetic. I think ( …

Let $X$ be a set of ordinals. If $X$ has no largest element, then \[ \sup X \notin X \subseteq \sup X, \] and $\sup X$ is the smallest ordinal $\alpha$ such that $X \subseteq \a …

Consider the theory $\mathrm{PA}+\mathrm{BHO}$ consisting of first-order Peano arithmetic ($\mathrm{PA}$) enriched by an axiom scheme which allows well-founded induction up to any …

I was wondering what the cardinality of $\omega\uparrow^\omega\omega$ is, with $\uparrow$ being Knuth's up-arrow notation. I ask this purely out of curiosity; after finding out abo …

Let $\alpha\in\mathbf{Ord}$ and $n\in\mathbb{N}^+$. Let $F_\alpha(n)$ be the number of distinct values taken by ordinal exponentiation $\underbrace{\alpha \hat{\phantom{\hat{}}} \ …

This is a question in two parts. Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to …

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulat …

While working on a project in operator algebras with a collaborator (and fellow MO user), we are able to successfully complete a transfinite induction assuming that the following h …

Define $\tau: \mathbf{Ord} \to \mathbf{Ord}$ such that $\tau(\alpha)$ is the order type of the minimal set $S$ of ordinals such that $\alpha \in S$ and $S$ is closed under ordinal …

The Cantor Normal Form Theorem states that every ordinal $\alpha > 0$ can be uniquely expressed in the form $$\omega^{\beta_1}k_1 + \omega^{\beta_2}k_2 + \dots + \omega^{\beta_n}k_ …

Hello, in a recent MO question, link, discussing the current foundations of mathematics, the author linked a video lecture by Prof. Voevodsky, which argues against the principle …

What is the least recursive ordinal $\alpha$ such that there is no algorithm in complexity class $\mathsf{P}$ which implements a well-ordering of $\mathbb{N}$ with order type $\alp …

Notation: $k, m, n$ are non-negative integers $f, g, h$ are functions $\mathbb{N} \to \mathbb{N}$ $f^k$ is $k$-th iterate of the function $f$: $f^0(n)=n, f^{k+1}(n)=f^k(f(n))$ $f …