[Paper Review] Saccharinity with ccc

Haim Horowitz, Saharon Shelah|arXiv (Cornell University)|Oct 9, 2016

Advanced Topology and Set Theory3 citations

TL;DR

This paper constructs Suslin ccc forcing notions using creature forcing technology to show that ZFC is equiconsistent with ZF + 'all sets of reals are $I_{\mathbb{Q},\aleph_1}$-measurable' + 'there exists an $\omega_1$-sequence of distinct reals', without requiring an inaccessible cardinal. The forcing notions are non-sweet and non-homogeneous, yet preserve ccc and ensure that all definable sets modulo the ideal are Borel, achieving a strong regularity property in a model violating $AC_{\aleph_0}$.

ABSTRACT

Using creature technology, we construct families of Suslin ccc non-sweet forcing notions $\mathbb Q$ such that $ZFC$ is equiconsistent with $ZF+$"every set of reals equals a Borel set modulo the $(\leq \aleph_1)$-closure of the null ideal associated with $\mathbb Q$"+"there is an $ω_1$-sequence of distinct reals". This answers a question of the second author and Kellner. As an application of independent interest, we also show how our forcing adds a new $Π^1_2$ singleton over $L$ without relying on $L$-combinatorics.

Motivation & Objective

• To classify definable ccc forcing notions by the consistency strength of ZF + 'all sets of reals are $I_{\mathbb{Q},\kappa}$-measurable' for $\kappa = \aleph_1$.
• To investigate whether such regularity properties can be achieved without assuming an inaccessible cardinal or the 'sweetness' property.
• To construct non-sweet, Suslin, ccc forcing notions that yield models where all sets of reals are $I_{\mathbb{Q},\aleph_1}$-measurable.
• To explore the role of non-wellfounded iterations and creature forcing in achieving homogeneity and regularity in models without full choice.

Proposed method

• Uses creature forcing techniques from [RoSh470] and [RoSh628] to define parameterized forcing notions $\mathbb{Q}_{\mathbb{n}}^1$ and $\mathbb{Q}_{\mathbb{n}}^2$ with finite-branching trees and norms on successors.
• Implements non-wellfounded linear orders in iterations to achieve strong homogeneity despite non-homogeneous individual forcing notions.
• Defines an ideal $I_{\mathbb{Q},\kappa}$ associated with a forcing notion $\mathbb{Q}$ adding a generic real, generalizing Lebesgue and Baire properties.
• Employs a notion of 'far' parameters to ensure independence between generic reals and control their distribution in the extension.
• Applies a compactness argument and proves that being a maximal antichain is a Borel property in $\mathbb{Q}_{\mathbb{n}}^2$, ensuring definability.
• Uses non-constructive choice principles in the ground model to obtain an $\omega_1$-sequence of distinct reals, preserved in the extension due to ccc.

Experimental results

Research questions

• RQ1 Can the consistency strength of 'all sets of reals are $I_{\mathbb{Q},\aleph_1}$-measurable' be reduced below that of an inaccessible cardinal for ccc forcing notions?
• RQ2 Is it possible to achieve such regularity without assuming the 'sweetness' property, especially in non-sweet ccc forcing?
• RQ3 Can a model of ZF + 'all sets of reals are $I_{\mathbb{Q},\aleph_1}$-measurable' + 'there is an $\omega_1$-sequence of distinct reals' be constructed without $AC_{\aleph_0}$?
• RQ4 What is the role of non-wellfounded iterations in achieving homogeneity in non-homogeneous forcing constructions?
• RQ5 Can similar results be extended to the ideal $I_{\mathbb{Q},\aleph_0}$, or is $\aleph_1$-based ideal essential?

Key findings

• The paper constructs Suslin ccc forcing notions $\mathbb{Q}_{\mathbb{n}}^i$ using creature forcing that are non-sweet and non-homogeneous, yet preserve ccc.
• It proves that ZFC is equiconsistent with ZF + 'all sets of reals are $I_{\mathbb{Q}_{\mathbb{n}}^i,\aleph_1}$-measurable' + 'there exists an $\omega_1$-sequence of distinct reals'.
• The model constructed satisfies ZF + 'all definable sets modulo $I_{\mathbb{Q}_{\mathbb{n}}^i,\aleph_1}$ are Borel' but fails $AC_{\aleph_0}$, indicating a finer consistency strength classification.
• The forcing notions ensure that every generic real is explicitly determined by the trunks of conditions, and no new reals are added beyond those in the iteration.
• The cofinality of the iteration order is $>\aleph_1$, which ensures that not all generic reals are covered by $\aleph_1$-many Borel sets from the ideal.
• The result shows that the existence of an $\omega_1$-sequence of distinct reals is consistent with strong regularity properties, even though it contradicts Lebesgue measurability of all sets of reals.