Subcomplete forcing principles and definable well‐orders

Fuchs, G.

doi:10.1002/malq.201800008

Cited by 4 publications

(2 citation statements)

References 30 publications

(96 reference statements)

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“…While my quest to deduce consequences of Martin's Maximum from SCFA (or some related forcing principles for subcomplete forcing) has been fairly successful in many respects, such as the failure of (weak) square principles and the reflection of stationary sets of ordinals [20], and even the existence of well-orders of P(ω 1 ) [19], it remained unclear until recently how to find an analog of SRP that relates to SCFA like SRP relates to Martin's Maximum. Thus, I was looking for a version of SRP that follows from SCFA and that, in turn, implies the major consequences of SCFA.…”

Section: Introductionmentioning

confidence: 99%

Canonical fragments of the strong reflection principle

Fuchs¹

2020

Preprint

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For an arbitrary forcing class Γ, the Γ-fragment of Todorčević's strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for Γ implies the Γ-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. Contents1. Introduction 1 2. Γ-projective stationarity and the Γ-fragment of SRP 3 2.1. Some background and motivation for SRP 3 2.2. Relativizing to a forcing class 6 2.3. The subcomplete fragment of SRP 7 3. Consequences 13 3.1. Barwise theory and a technical lemma 13 3.2. Friedman's problem, the failure of square, and SCH 21 3.3. Mutual stationarity 26 4. Limitations and separations 31 4.1. The general setting 31 4.2. The CH setting: a less canonical separation 35 4.3. The CH setting: a canonical separation at ω 2 37 5. Questions 46 References 47

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Section: Introductionmentioning

confidence: 99%

Canonical fragments of the strong reflection principle

Fuchs¹

2020

Preprint

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“…Namely, it is known to be consistent that CH holds and every Aronszajn tree is special (see [18]). But in a model of this theory, any ccc forcing preserves Aronszajn trees of height ω 1 and any width (see the proof of [9,Theorem 4.23]), while the bounded forcing axiom for any forcing that adds a real fails (since it implies the failure of CH; see [6,Obs. 4…”

Section: Introductionmentioning

confidence: 99%

Aronszajn tree preservation and bounded forcing axioms

Fuchs

2020

Preprint

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I investigate the relationships between three hierarchies of reflection principles for a forcing class Γ: the hierarchy of bounded forcing axioms, of Σ 1 1 -absoluteness and of Aronszajn tree preservation principles. The latter principle at level κ says that whenever T is a tree of height ω 1 and width κ that does not have a branch of order type ω 1 , and whenever P is a forcing notion in Γ, then it is not the case that P forces that T has such a branch. Σ 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 ω are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don't add reals.

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Aronszajn Tree Preservation and Bounded Forcing Axioms

Fuchs¹

2021

J. symb. log.

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I investigate the relationships between three hierarchies of reflection principles for a forcing class : the hierarchy of bounded forcing axioms, of -absoluteness, and of Aronszajn tree preservation principles. The latter principle at level says that whenever T is a tree of height and width that does not have a branch of order type , and whenever is a forcing notion in , then it is not the case that forces that T has such a branch. -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 are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don’t add reals.

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Subcomplete forcing principles and definable well‐orders

Cited by 4 publications

References 30 publications

Canonical fragments of the strong reflection principle

Canonical fragments of the strong reflection principle

Aronszajn tree preservation and bounded forcing axioms

Aronszajn Tree Preservation and Bounded Forcing Axioms

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