Questions tagged [recursive-ordinals]

This seems like it should be a pretty well-studied question but I can't seem to find an easy answer: Is the theory $(\varepsilon_0, +, \omega^{\ \cdot}, 0, 1)$ decidable? From Is the theory of $(\...

Consider $\omega^2$. We can build a simple bijection between the ordinal and $\omega$ similarly to how the bijection between $\mathbb{Q}$ and $\mathbb{N}$ can be built. I was wondering if there is a ...

In Harrington's mimeographed notes (see here) solving McLaughlin's conjecture he builds reals $f \in \omega^\omega$ which have the property of being $\alpha$-subgeneric defined as follows. He does ...

The notion of a “computable pseudo-ordinal”, i.e. a computable linear order with no hyperarithmetical descending chains, is an old one going back to Stephen Kleene. Joe Harrison wrote the definitive ...

Given a perfect tree $T$ on $2^{<\omega}$ viewed as a function from $2^{<\omega}$ to $2^{<\omega}$ define the measure of a subset of $[T]$ to be the measure of it's preimage under the usual ...

What is the complexity (e.g. is it $\Sigma^0_1$, arithmetic, fully $\Pi^1_1$) of the relation $|a| < |b|$ given two notations $a, b \in \mathscr{O}$ (Kleene's O)? What about the case where only one ...