相关论文: GCH implies AC, a Metamath Formalization
We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.
We provide a short proof of Shelah's eventual categoricity conjecture, assuming the Generalized Continuum Hypothesis ($GCH$), for abstract elementary classes (AEC's) with interpolation, a strengthening of amalgamation which is a necessary…
There are several extensions of the classical Banach Fixed Point Theorem in technical literature. A branch of generalizations replaces usual contractivity by weaker but still effective assumptions. Our note follows this stream, presenting…
We introduce a notion of the ``explanation" of one (generalized) probabilistic model by another as particular kind of span in the category $\Prob$ of probabilistic models and morphisms. We show that explanations compose under a standard…
The Axiom of Choice (AC for short) is the most (in)famous axiom of the usual foundations of mathematics, ZFC set theory. The (non-)essential use of AC in mathematics has been well-studied and thoroughly classified. Now, fragments of…
Generalized Higman's Theorem is the direct counterpart of Higman's Theorem that asserts the closure of the class of \emph{better} quasi-orders, instead of the class of \emph{well} quasi-orders, under the construction $P\mapsto P^{<\omega}$…
We argue that we solved Hilbert's first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let lambda to the revised power of kappa, denoted lambda^{[kappa]}, be the minimal…
Czachor's recent proposal introduces a form of non-Newtonian calculus built by pulling back arithmetic operations through arbitrary bijections between continua. Although the idea is mathematically inventive, it runs into serious conceptual…
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum…
We deal with relatives of GCH which are provable. In particular we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice are assumed but some of the results gives…
The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the…
We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying…
In this paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions,…
Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary…
In this note we introduce a notion of a generically (strongly generically) NP-complete problem and show that the randomized bounded version of the halting problem is strongly generically NP-complete.
There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly…
In this paper, we describe the formalization of the axiom of choice and several of its famous equivalent theorems in Morse-Kelley set theory. These theorems include Tukey's lemma, the Hausdorff maximal principle, the maximal principle,…
In this paper we present a proof of Goodman's Theorem, a classical result in the metamathematics of constructivism, which states that the addition of the axiom of choice to Heyting arithmetic in finite types does not increase the collection…
We define a generalization of Chacon's classical automorphism and answer the question of whether its important properties remain. We calculate the family of polynimials representing the automorphism, given in recurrence formulae, and infer…