Related papers: Proof-theoretic dilator and intermediate pointclas…
We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their…
Schmerl and Beklemishev's work on iterated reflection achieves two aims: It introduces the important notion of $\Pi^0_1$-ordinal, characterizing the $\Pi^0_1$-theorems of a theory in terms of transfinite iterations of consistency; and it…
In this paper we give an ordinal analysis of a set theory with $\Pi_{1}$-Collection.
A predilator is a particularly uniform transformation of linear orders. We have a dilator when the transformation preserves well-foundedness. Over the theory $\mathsf{ACA}_0$ from reverse mathematics, any $\Pi^1_2$-formula is equivalent to…
We describe a proof-theoretic bound on $Sigma_{2}$-definable countable ordinals in Kripke-Platek set theory with $Pi_{1}$-Collection and the existence of $omega_{1}$.
A way of testing the $\pi \pi$ predictions of Chiral Perturbation Theory against experimental data is to use dispersion relations to continue experimental information into the subthreshold region where the theory should unambiguously apply.…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
In this paper we expound some basic ideas of proof theory for theories of ordinals such that there are many stable ordinals below the ordinals.
In this paper we give an ordinal analysis of a set theory with $\Pi_{N}$-Collection.
In this note we provide a quick proof of the Sklar's Theorem on the existence of copulas by using the generalized inverse functions as in the one dimensional case, but a little more sophisticated.
Proofs are traditionally syntactic, inductively generated objects. This paper reformulates first-order logic (predicate calculus) with proofs which are graph-theoretic rather than syntactic. It defines a combinatorial proof of a formula…
This chapter provides a comprehensive overview of proof-theoretic methods for establishing interpolation properties across a range of logics, including classical, intuitionistic, modal, and substructural logics. Central to the discussion…
We investigate the cyclic proof theory of extensions of Peano Arithmetic by (finitely iterated) inductive definitions. Such theories are essential to proof theoretic analyses of certain `impredicative' theories; moreover, our cyclic systems…
Fast-growing hierarchies are sequences of functions obtained through various processes similar to the ones that yield multiplication from addition, exponentiation from multiplication, etc. We observe that fast-growing hierarchies can be…
The explicit constructions of minimal isometric, and minimal unitary dilations of an arbitrary linear pencil of operators $T(\lambda)=T_0+\lambda T_1$ consisting of contractions on a separable Hilbert space for $|\lambda |=1$, which…
For $0<r<1$, let us consider the following annulus: \[ \mathbb A_r= \{ z\in \mathbb C\, : \, r<|z|<1 \}. \] A Hilbert space operator $T$ for which $\overline{\mathbb A}_r$ is a spectral set is called an $\mathbb A_r$-\textit{contraction}.…
There is some disagreement on whether Likert scale data should be treated as ordinal or continuous. This paper treats Likert data as ordinal, uses non-parametric hypothesis testing, and clustering to validate those variables that have…
This paper is a prelude and elaboration on Proofs that Modify Proofs. Here we present an ordinal analysis of a fragment of the $\mu$-calculus around the strength of parameter-free $\Pi^1_2$-comprehension using the same approach as that…
We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original…
In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with…