Related papers: Some Natural Internal Forcing Schemata Extending Z…

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and…

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the…

If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms introduced by Hamkins.

We isolate a combinatorial property of capacities leading to a construction of proper forcings. Then we show that many classical capacities such as the Newtonian capacity satisfy the property.

This is a report on state-of-the-art on the question of developing higher analogues of the forcing axiom PFA. Recently there have been several attempts to develop forcing axioms analogous to the proper forcing axiom (PFA) for cardinals of…

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to…

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Lof's type theory (hence…

In this paper we prove that the maximum principle in forcing is equivalent to the axiom of choice. The maximum principle is the property of forcing: p ||- exists x theta(x) iff for some name tau p ||- theta(tau). We also look at three…

We consider the open Toda chain with external forcing, and in the case when the forcing stretches the system, we derive the longtime behavior of solutions of the chain. Using an observation of J\"{u}rgen Moser, we then show that the system…

We present a system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK…

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…

This is an overview about a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along $\omega_1$ is given. Then its direct limit satisfies ccc by…

We introduce the forcing model of IZFA (Intuitionistic Zermelo-Fraenkel set theory with Atoms) for every Grothendieck topology and prove that the topos of sheaves on every site is equivalent to the category of 'sets in this forcing model'.

We ask for a given system of polynomials f_1,...,f_n and f over the complex numbers when there exist continuous functions q_1,...,q_n such that q_1 f_1+...+q_n f_n = f. This condition defines the continuous closure of an ideal. We give…

This paper considers the phenomenon of distinct regional frequencies recently observed in some power systems. First, a reduced-order mathematical model describing this behaviour is developed. Then, techniques to solve the model are…

We prove it consistent relative to ZFC that all nontrivial forcings of size $\aleph _1$ add a Cohen real.

We study how to infer new choices from previous choices in a conservative manner. To make such inferences, we use the theory of choice functions: a unifying mathematical framework for conservative decision making that allows one to impose…

We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present…

We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([omega]^{aleph_0},supseteq^*) is proper. They cover many reasonable forcing notions.

Using a variation of Woodin's $\mathbb{P}_{\mathrm{max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $\omega_{2}$ consisting of functions…