Every good introduction to set-theoretic forcing starts with basic forcing algebras like ”adding a Cohen real” or ”adding a Cohen subset of ω1”. It seems that these algebras are particularly simple mainly because they contain a dense subtree, so we might as well pretend we are forcing with a tree. Such a forcing is especially easy to visualize as the generic object will be a branch through the ...

For a boundary-value problem of an ordinary differential equation, we prove that the anti-maximum principle holds when the forcing term satisfies an integral inequality. As applications, we consider linear and nonlinear models arising from fishery management problems.

The Maximality Principle mp is a scheme which states that if a sentence of the language of zfc is true in some forcing extension V , and remains true in any further forcing extension of V , then it is true in all forcing extensions of V . A modified maximality principle mpΓ arises when considering forcing with a particular class Γ of forcing notions. A parametrized form of such a principle, mpΓ...

Posner and Robinson [4] proved that if S ⊆ ω is non-computable, then there exists a G ⊆ ω such that S ⊕ G ≥T G′. Shore and Slaman [7] extended this result to all n ∈ ω, by showing that if S T ∅(n−1) then there exists a G such that S ⊕ G ≥T G(n). Their argument employs KumabeSlaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing i...

A. We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal. 1. I In the seminal paper [10] Solovay proved that in the Levy model (after collap...

In this note, we give a survey on recent developments pertaining to forcing axioms with emphasis on their characterizations and consequences connected to the Continuum Problem. Nonspecialists in set theory and/or students in mathematics are supposed to be the readers, and thus we tried to make this as self-contained as possible. Due to the limitation on the extent of the article, however, many ...

This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conserv...

Azriel Levy (1934–) did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcin...

‎in this paper‎, ‎we investigate a damped korteweg-de‎ ‎vries equation with forcing on a periodic domain‎ ‎$mathbb{t}=mathbb{r}/(2pimathbb{z})$‎. ‎we can obtain that if the‎ ‎forcing is periodic with small amplitude‎, ‎then the solution becomes‎ ‎eventually time-periodic.

using the notion of anti fuzzy points and its besideness to and nonquasi-coincidence with a fuzzy set, new concepts in anti fuzzy subalgebras in bck/bci-algebras are introduced and their properties and relationships are investigated.