47 problems

A type-two well-ordering principle is a well-ordering principle whose transformation of linear orders is of type two, together with the assertion that the resulting order is well-f…

Conservativity-spectrum conjecture. In general, conservativity spectra consist precisely of $\ell$-sequences.

For each $n$, let $a,b<2^{n-1}$ satisfy $p_n(a)=p_n(b)=2^k$. Let $c$ be the least element such that $\gamma_k$ is in the range of $c$, and let $c\gamma_i=\gamma_k$. Assume that…

For each $n$, let $A_n$ be the cyclic algebra considered in the paper. If $a<2^n-1$, write $p_n(a)=2^k$, and if $c+1$ is the threshold of $a$ in $A_n$, require that…

Let $\Gamma$ be the set of critical points of the embeddings in $\Cal A$, and let $\Sigma$ be the closure of $\Gamma$ under $a"\alpha=\sup\{a\xi:\xi<\alpha\}$ for $a\in\Cal A$. Let…

Ordinal-height conjecture. The predicatively recursive functions on a well-founded partial order correspond to the predicatively recursive functions on its associated ordinal.

Let $(A_n)_{n\geq 1}$ be the Aurellion function sequence defined recursively in the paper, and let $f_\alpha$ denote the fast-growing hierarchy indexed by ordinals $\alpha$. Aurell…

Turing's conjecture. For every arithmetical theorem $\varphi$, there is a computable presentation of an ordinal $\alpha$ such that

In second-order arithmetic, let $\textsf{RCA}_0$ and $\Pi^1_1\textsf{-CA}_0$ denote the indicated subsystems, and consider the theorem that inverse Goodstein sequences terminate, w…

Isomorphism conjecture. There is an isomorphism

Let $gamma$ be an ordinal with Cantor normal form, and write $alpha_1$ for its leading exponent. Let $beta(gamma)$ be the corresponding beta space, $sf L(beta(gamma))$ its logic, a…

Let $S_u$ for $u\in\{0,1,2\}$ be the sets … ordered by $x\preccurlyeq y$ if and only if $x>y$. For a well-ordered set $L$ and an ordinal $\alpha$, let $L[\alpha]$ denote the elemen…

In infinite chess, a position has a game value given by the ordinal rank measuring how many moves a player can force before winning, when such a value is defined. Evans–Hamkins con…

The alternative normal forms in Example give rise to a terminating Goodstein process, but the methods in the paper do not establish lower bounds on such processes. Alternative Good…

Let $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and $(c_\alpha)_{\alpha<\xi}$ be the transfinite sequences associated with the three complexity classes modulo $3$. Block-…

Let $\|n\|$ denote the complexity of a natural number, and let $\xi$ be an ordinal. Consider three transfinite sequences $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and…

Arias's conjecture. The set $A$ has finite symmetric difference with $E$, and the one-sided difference $A\setminus E$ is a finite subset of

Let $\mathcal{F}$ denote the set of fusible numbers, and let $\mathcal{F}(\alpha)$ denote its value at the ordinal $\alpha$. Fusible-number self-similarity conjecture. The fusible…

Canonicity conjecture. If one had a convincing definition of natural, then for any reasonable theory $T$ there would be a natural presentation $\prec$ of a recursive ordinal such t…

Relativized patterns conjecture. A relativization of patterns of resemblance to dilators should be equivalent to $\Pi^1_1$-comprehension.

Let $T$ be a strong system with an ordinal analysis based on ordinal representation systems $\mathsf{OT}$. These systems give rise to functors … that send dilators to ordinal repre…

Interval representation conjecture. Every fusible number in $I_{x,n}$, for $n\geq 1$, can be written as

Closure-ordinal conjecture. The closure ordinal for this system is less than or equal to $\omega^\omega$.

For a linear order $L$, write $L^{\star}$ for its reverse, and let $\leq_c$ denote computable embeddability between pairs of structures. The preceding results establish computable…

The paper considers programs that notate increasingly large ordinals, including constructions reaching beyond previously developed techniques. Intelligence-growth conjecture. Each…