Related papers: Predicativity through transfinite reflection
Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible…
Periodic multilayers of various periods were prepared according to an algorithm proposed by the authors. The reflectivity properties of these systems were investigated using neutron reflectometry.The obtained experimental results were…
In this paper the lightface $\Pi^{1}_{1}$-Comprehension axiom is shown to be proof-theoretically strong even over $\mbox{RCA}_{0}^{*}$, and we calibrate the proof-theoretic ordinals of weak fragments of the theory $\mbox{ID}_{1}$ of…
Gentzen's 1936 proof of the consistency of Peano Arithmetic was a significant result in the foundations of mathematics. We provide here a modified version of the proof, based on G\"{o}del's reformulation, and including additional details…
The classes of relativized relation algebras (whose units are not necessarily transitive as binary relations) are known to be finitely axiomatizable. In this article, we give a new proof for this fact that is easier and more transparent…
Classically, any structure for a signature $\Sigma$ may be completed to a model of a desired regular theory $T$ by means of the chase construction or small object argument. Moreover, this exhibits $\mathrm{Mod}(T)$ as weakly reflective in…
Physical reasoning requires forward prediction: the ability to forecast what will happen next given some initial world state. We study the performance of state-of-the-art forward-prediction models in the complex physical-reasoning tasks of…
We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant `true' by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform…
We define reflective numbers and their iterative summations. We provide classification of reflective numbers based on their iterative cyclical limits.
Linear arithmetics are extensions of Presburger arithmetic (Pr) by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages. In this…
He et al. (2018) have called into question the utility of pre-training by showing that training from scratch can often yield similar performance to pre-training. We show that although pre-training may not improve performance on traditional…
In this work, we study some novel applications of conformal inference techniques to the problem of providing machine learning procedures with more transparent, accurate, and practical performance guarantees. We provide a natural extension…
The success of neural networks comes hand in hand with a desire for more interpretability. We focus on text classifiers and make them more interpretable by having them provide a justification, a rationale, for their predictions. We approach…
A method is given for determining a mixed social choice out of a paired-comparison matrix. The method combines a projection procedure introduced in previous papers of the same authors and a classical method due to Zermelo. The resulting…
This article proposes predictive economics as a distinct analytical perspective within economics, grounded in machine learning and centred on predictive accuracy rather than causal identification. Drawing on the instrumentalist tradition…
In this note, we investigate iterations of consistency, local and uniform reflection over $\mathbf{HA}$ (Heyting Arithmetic). In the case of uniform reflection, we give a new proof of Dragalin's extension of Feferman's completeness theorem…
In a recent paper, Enayat and Le lyk [2024] show that second order arithmetic and countable set theory are not definitionally equivalent. It is well known that these theories are biinterpretable. Thus, we have a pair of natural theories…
If M is a nonstandard model of Peano Arithmetic, then M is lofty iff M has a simple elementary extension that is recursively saturated. This had previously been known for countable M.
Inspired by Leivant's work on absolute predicativism, Bellantoni and Cook in 1992 introduced a structurally restricted form of recursion called predicative recursion. Using this recursion scheme on the inductive structures of natural…
Deep neural networks exhibit remarkable performance, yet their black-box nature limits their utility in fields like healthcare where interpretability is crucial. Existing explainability approaches often sacrifice accuracy and lack…