Related papers: Class forcing, the forcing theorem and Boolean com…
Relational verification encompasses research directions such as reasoning about data abstraction, reasoning about security and privacy, secure compilation, and functional specificaton of tensor programs, among others. Several relational…
We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a…
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which…
With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for…
This note addresses the continuum problem, taking advantage of the breakthrough mentioned in the subtitle, and relating it to many recent advances occurring in set theory.
Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on…
We show that it is consistent with MA + the negation of CH, that the Forcing Axiom fails for all forcing notions in the class of omega^omega-bounding forcing notions with norms of "Norms on possibilities I: forcing with trees and…
As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently…
Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize…
In a recent article by Farah and the authors, a strong lifting theorem was proved for a class of coordinate-respecting maps between reduced products of discrete structures, hereby working under mild Forcing Axioms. We generalise this…
We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable $C^*$-algebras with the metric approximation property…
A class of languages C is perfect if it is closed under Boolean operations and the emptiness problem is decidable. Perfect language classes are the basis for the automata-theoretic approach to model checking: a system is correct if the…
In this short note, we shall prove some observations regarding the connection between indestructible $\omega_1$-guessing models and the $\omega_1$-approximation property of forcing notions.
We prove that if $\mathcal{A}$ is a $\sigma$-complete Boolean algebra in a model $V$ of set theory and $\mathbb{P}\in V$ is a proper forcing with the Laver property preserving the ground model reals non-meager, then every pointwise…
In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored.…
We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{\omega_1}+$all sets of reals are $I_{\mathbb Q,\aleph_0}$-measurable'' implies the existence of an inner model with a measurable…
The independence of the continuum hypothesis is a result of broad impact: it settles a basic question regarding the nature of N and R, two of the most familiar mathematical structures; it introduces the method of forcing that has become the…
A set $A$ of integers is called total if there is an algorithm which, given an enumeration of $A$, enumerates the complement of $A$, and called cototal if there is an algorithm which, given an enumeration of the complement of $A$,…
The analysis of theory-confirmation generally takes the deductive form: show that a theory in conjunction with physical data and auxiliary hypotheses yield a prediction about phenomena; verify the prediction; provide a quantitative measure…
We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC}+\neg\mathit{CH}$ and $\mathit{ZFC}+\mathit{CH}$.…