arXiv subjects

Gunter Fuchs

Publications and source records attributed to Gunter Fuchs.

17 recordsLinked to original sources

arXiv2021

The Diagonal Strong Reflection Principle and its fragments

A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated to arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle are analyzed, and consequences are presented. Some of these consequences are ``exact'' versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

math.LO↗

arXiv2020

Canonical fragments of the strong reflection principle

For an arbitrary forcing class $\Gamma$, the $\Gamma$-fragment of Todorcevic's strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for $\Gamma$ implies the $\Gamma$-fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their corresponding forcing axioms are established.

math.LO↗

arXiv2020

Iteration theorems for subversions of forcing classes

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega\omega$-bounding forcing notions, 2) the class of subproper, $T$-preserving forcing notions (where $T$ is a fixed Souslin tree) and 3) the class of subproper, $[T]$-preserving forcing notions (where $T$ is an $\omega_1$-tree) are iterable with revised countable support. In the second part, we adopt Miyamoto's theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega_1$, and, in the case of subcompleteness, don't add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

math.LO↗

arXiv2020

Separating diagonal stationary reflection principles

We introduce three families of diagonal reflection principles for matrices of stationary sets of ordinals. We analyze both their relationships among themselves and their relationships with other known principles of simultaneous stationary reflection, the strong reflection principle, and the existence of square sequences.

math.LO↗

arXiv2020

Aronszajn tree preservation and bounded forcing axioms

I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma$: the hierarchy of bounded forcing axioms, of $\Sigma^1_1$-absoluteness and of Aronszajn tree preservation principles. The latter principle at level $\kappa$ says that whenever $T$ is a tree of height $\omega_1$ and width $\kappa$ that does not have a branch of order type $\omega_1$, and whenever $P$ is a forcing notion in $\Gamma$, then it is not the case that $P$ forces that $T$ has such a branch. $\Sigma^1_1$-absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don't add reals, the three principles at level $2^\omega$ are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don't add reals.

math.LO↗

arXiv2017

Weak square and stationary reflection

It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{\mathrm{cf}(\lambda)} < \lambda$ for all $\mu < \lambda$, then $\square^*_\lambda$ entails the existence of a non-reflecting stationary subset of $E^{\lambda^+}_{\mathrm{cf}(\lambda)}$ in the forcing extension for adding a single Cohen subset of $\lambda^+$. It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of $\square^*_\lambda$ for every singular cardinal $\lambda$ of countable cofinality.

math.LO↗

arXiv2017

Subcomplete forcing principles and definable well-orders

It is shown that the boldface maximality principle for subcomplete forcing, together with the assumption that the universe has only set-many grounds, implies the existence of a (parameter-free) definable well-ordering of $\mathcal{P}(\omega_1)$. The same conclusion follows from the boldface maximality for subcomplete forcing, assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that $x^\#$ does not exist, for some $x\subseteq\omega_1$, implies the existence of a well-order of $\mathcal{P}(\omega_1)$ which is $\Delta_1$-definable without parameters, and $\Delta_1(H_{\omega_2})$-definable using a subset of $\omega_1$ as a parameter. This well-order is in $L(\mathcal{P}(\omega_1))$. Enhanced version of bounded forcing axioms are introduced that are strong enough to have the implications of the maximality principles mentioned above.

math.LO↗

arXiv2017

Subcomplete forcing, trees and generic absoluteness

We investigate properties of trees of height $\omega_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $\omega_1$-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width $\omega_1$. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width $\omega_1$ and generic absoluteness of $\Sigma^1_1$-statements over first order structures of size $\omega_1$, also for other canonical classes of forcing.

math.LO↗

arXiv2017

Inner-model reflection principles

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the L\'evy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $\Pi_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.

math.LO↗

arXiv2017

Boolean ultrapowers, the Bukovsky-Dehornoy phenomenon, and iterated ultrapowers

We show that while the length $\omega$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $\omega$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Prikry forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct limit model by the critical sequence as the Bukovsky-Dehornoy phenomenon, and we develop a sufficient criterion (the existence of a simple skeleton) for when a version of this phenomenon holds in the context of Boolean ultrapowers. Assuming that the canonical generic filter over the Boolean ultrapower model has what we call a continuous representation, we show that the Boolean model consists precisely of those members of the intersection model that have continuously and eventually uniformly represented codes.

math.LO↗

arXiv2017

The strong Prikry property

I isolate a combinatorial property of a poset $\mathbb{P}$ that I call the strong Prikry property, which implies the existence of an ultrafilter on the complete Boolean algebra $\mathbb{B}$ of $\mathbb{P}$ such that one inclusion of the Boolean ultrapower version of the so-called \Bukovsky-Dehornoy phenomenon holds with respect to $\mathbb{B}$ and $U$. I show that in all cases that were previously studied, and for which it was shown that they come with a canonical iterated ultrapower construction whose limit can be described as a single Boolean ultrapower, the posets in question satisfy this property: Prikry forcing, Magidor forcing and generalized Prikry forcing.

math.LO↗

arXiv2017

Split Principles

We introduce the split principles and show that they bear tight connections to large cardinal properties such as inaccessibility, weak compactness, subtlety, almost ineffability and ineffability, as well as classical combinatorial objects such as Aronszajn trees, Souslin trees or square principles. We exhibit correspondences between certain split principles and splitting numbers at uncountable cardinals.

math.LO↗

arXiv2015

Ehrenfeucht's lemma in set theory

Ehrenfeucht's lemma (1973) asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht's lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht's lemma holds fully for models of set theory satisfying $V=HOD$. We show that the lemma can fail, however, in models of set theory with $V\neq HOD$, and it necessarily fails in the forcing extension to add a generic Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht's lemma, namely, the principles of the form $EL(A,P,Q)$, which asserts that whenever an object $b$ is definable from some $a\in A$ using parameters in $P$, with $b\neq a$, then the types of $a$ and $b$ over $Q$ are different. We also consider various analogues of Ehrenfeucht's lemma obtained by using algebraicity in place of definability, where a set $b$ is algebraic in $a$ if it is a member of a finite set definable from $a$ (as in Hamkins, Leahy arXiv:1305.5953). Ehrenfeucht's lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using similar analysis, we answer two open questions posed by Hamkins and Leahy, by showing that (i) algebraicity and definability need not coincide in models of set theory and (ii) the internal and external notions of being ordinal algebraic need not coincide.

math.LO↗

arXiv2015

Incomparable $\omega_1$-like models of set theory

We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $\omega_1$-like models of set theory. Specifically, under the $\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{\omega_1}$ many $\omega_1$-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $\omega_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $\omega_1$-like model of PA whose structure of hereditarily finite sets is not universal for the $\omega_1$-like models of set theory.

math.LO↗

arXiv2011

Set-Theoretic Geology

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.

math.LO↗

arXiv2007

Changing the Heights of Automorphism Towers by Forcing with Souslin Trees over L

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

math.LO↗

arXiv2006

Degrees of rigidity for Souslin trees

We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.

math.LO↗