Related papers: On Recurrence Axioms
In classical analysis, the convergence behavior of power series solutions to differential or recurrence equations is generally assumed to be invariant under internal rearrangement. This paper challenges that belief by proving that, for…
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
Despite the success of test-time scaling, Large Reasoning Models (LRMs) frequently encounter repetitive loops that lead to computational waste and inference failure. In this paper, we identify a distinct failure mode termed Circular…
Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $\kappa$---which we denote $\Diamond_{\text{Lav}}(\kappa)$---asserting the existence of a single function from $\kappa \to H_\kappa$ that behaves much like a…
There is an emerging interest in generating robust counterfactual explanations that would remain valid if the model is updated or changed even slightly. Towards finding robust counterfactuals, existing literature often assumes that the…
The compactness lemma in programming language theory states that any recursive function can be simulated by a finite unrolling of the function. One important use case it has is in the logical relations proof technique for proving properties…
I discuss classical and quantum recurrence theorems in a unified manner, treating both as generalisations of the fact that a system with a finite state space only has so many places to go. Along the way I prove versions of the recurrence…
Assume ZFC. Let $\kappa$ be a cardinal. A ${<\kappa}$-ground is a transitive proper class $W$ modelling ZFC and such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<\kappa}$. The $\kappa$-mantle is…
In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2 and that…
A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset P with no maximal element, there is a ccc forcing extension in which…
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping…
In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset…
The $p$-adic Littlewood Conjecture due to De Mathan and Teuli\'e asserts that for any prime number $p$ and any real number $\alpha$, the equation $$\inf_{|m|\ge 1} |m|\cdot |m|_p\cdot |\langle m\alpha \rangle|\, =\, 0 $$ holds. Here, $|m|$…
The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver's theorem and Bukovsk\'y's theorem assert that set-generic extensions of a given…
We introduce a variant of Martin's axiom, called the grounded Martin's axiom, which asserts that the universe is a ccc forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of…
Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels…
Let $(X,T,\mu,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there…
Dependence on the parameter is continuous when perturbations of the parameter preserves strict preference for one alternative over another. We characterise this property via a utility function over alternatives that depends continuously on…
This paper reports a modified axiomatic foundation of the analytic hierarchy process (AHP), where the reciprocal property of paired comparisons is broken. The novel concept of reciprocal symmetry breaking is proposed to characterize the…
We give an evidence of the Big Fix. The theory of wormholes and multiverse suggests that the parameters of the Standard Model are fixed in such a way that the total entropy at the late stage of the universe is maximized, which we call the…