Related papers: More on forcing iteration
By operations on models we show how to relate completeness with respect to permissive-nominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal…
We note that some form of the condition "$p_1, p_2$ have a $\leq_{\mathbb{Q}}$-lub in $\mathbb{Q}$" is necessary in some forcing axiom for $\lambda$-complete $\mu^+$-c.c. forcing notions. We also show some versions are really stronger than…
Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete Boolean algebra in $M.$ (In general a proper class.) We define a generalized notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing extension of…
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are…
In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic…
Suppose $\kappa$ is a singular strong limit cardinal of countable cofinality and let $\langle \kappa_{n}: n<\omega \rangle$ be an incrasing sequence of regular cardinals cofinal in $\kappa$. We show that if $cf(2^\kappa)= \kappa^+$, then…
We prove a compactness theorem for pseudopower operations of the form $pp_{\Gamma(\mu,\sigma)}(\mu)$ where $\aleph_0<\sigma=cf(\sigma)\leq cf(\mu)$. Our main tool is a result that has Shelah's cov vs. pp Theorem as a consequence. We also…
In a nutshell, we intend to extend Schoenberg's classical theorem connecting conditionally positive semidefinite functions $F\colon \mathbb{R}^n \to \mathbb{C}$, $n \in \mathbb{N}$, and their positive semidefinite exponentials $\exp(tF)$,…
We use high girth, high chromatic number hypergraphs to show that there are finite models of the equational theory of the semiring of nonnegative integers whose equational theory has no finite axiomatisation, and show this also holds if…
We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal $\alpha$ there exists an ordinal $\beta$ such that $1+\beta\cdot(\beta+\alpha)$ (ordinal arithmetic) admits an…
We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD,…
In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of…
We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not…
We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo…
In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear…
We study principles of the form: if a name $\sigma$ is forced to have a certain property $\varphi$, then there is a ground model filter $g$ such that $\sigma^g$ satisfies $\varphi$. We prove a general correspondence connecting these name…
Let M be a factor of type III with separable predual and with normal states phi_1,...,phi_k, omega with omega faithful. Let A be a finite dimensional C*-subalgebra of M. Then it is shown that there is a unitary operator u in M such that…
The aim of these lectures is to give a short introduction to forcing. We will avoid metamathematical issues as much as possible and similarly we will avoid performing the actual construction of forcing. We assume familiarity with basic…
We examine the existence (and mostly non-existence) of fresh sets in commonly used iterations of Prikry type forcing notions. Results of [4] are generalized. As an application, a question of a referee of [9] is answered. In addition…
Topological objects resulting from symmetry breakdown may be either stable or metastable depending on the pattern of symmetry breaking. However, if they trap zero-energy modes of fermions, and in the process acquire non-integer fermionic…