Long Strong Chains of Subsets of $ω_1$

Abstract: We force the existence of a chain of length $ω_3$ in $[ω_1]<sup>{ω_1}$ increasing modulo finite. The construction involves symmetric systems of models of two types as side conditions, introduced by the second author. This improves previous results of Koszmider and Veličković-Venturi.

• The paper introduces a forcing method using symmetric systems of two types to construct a strong ω₃-chain in [ω₁]^{ω₁} modulo finite.
• It demonstrates that the forcing is proper and ℵ₃-Knaster, ensuring cardinal preservation with effective amalgamation and monotonicity conditions.
• The findings extend previous work by closing the gap between known ω₂-chain results and higher-chain constructions, with implications for similar forcing applications.

Forcing Long Strong Chains in $[\omega_1]^{\omega_1}$

Context and Motivation

The combinatorial study of the space $[\omega_1]^{\omega_1}$, particularly sequences of uncountable subsets of $\omega_1$ ordered by inclusion modulo finite, has played a central role in higher set theory, especially with respect to cardinal invariants and the structure of the continuum at large cardinalities. Classical results, such as Baumgartner's construction of strong almost disjoint families of size arbitrarily large (modulo cardinal arithmetic constraints), provided the foundation for further questions about the possible lengths and structure of strong chains of subsets of $\omega_1$ and their relations to analogous chains of functions in $^{\omega_1}\omega_1$.

A key question is: How long can such strong chains of subsets exist, consistently with standard set-theoretical hypotheses? Previous results by Koszmider established the consistency of strong $\omega_2$-chains in $[\omega_1]^{\omega_1}$ using sophisticated proper forcing arguments with morass-based side conditions. However, extending these techniques to higher analogues (e.g., to $\omega_3$-long chains) appeared blocked by substantial combinatorial obstacles, highlighted by impossibility results of Shelah and Inamdar for even larger uncountable cardinals.

This paper addresses whether it is possible, under GCH, to construct in a cardinal-preserving way a chain of length $\omega_3$ in $[\omega_1]^{\omega_1}$, increasing modulo finite sets. The answer, in the affirmative, is achieved via refined forcing with symmetric systems of models of two types as side conditions, pushing the combinatorial technology beyond previous candidates and, crucially, not requiring models of size $[\omega_1]^{\omega_1}$0 or side conditions of three types.

Definitions and Preliminaries

The central objects under investigation are:

• Strong $[\omega_1]^{\omega_1}$1-Chain of Subsets of $[\omega_1]^{\omega_1}$2: A sequence $[\omega_1]^{\omega_1}$3 of subsets of $[\omega_1]^{\omega_1}$4 such that for all $[\omega_1]^{\omega_1}$5,
  • $[\omega_1]^{\omega_1}$6,
  • $[\omega_1]^{\omega_1}$7.
• Forcing with Symmetric Systems of Side Conditions: The authors use side conditions composed of countable (“small”) and $[\omega_1]^{\omega_1}$8-sized (“large”) elementary submodels, called $[\omega_1]^{\omega_1}$9-symmetric systems. These are closed under isomorphisms at the appropriate ($\omega_1$0) level and satisfy structural properties ensuring the desired preservation and amalgamation features for the forcing.

The paper assumes GCH throughout, which provides the combinatorial foundation, specifically guaranteeing that $\omega_1$1 and enabling the $\omega_1$2-Knaster property for the poset. The side conditions and associated amalgamation lemmas are drawn from previous work on symmetric systems (“Forcing with Symmetric Systems of Models of Two Types” [(Rodríguez, 2022)v2]).

Forcing Construction

The main technical achievement is the definition of a forcing poset $\omega_1$3 whose conditions encode finite approximations to the chain:

• Conditions: Each $\omega_1$4 consists of a finite $\omega_1$5-symmetric system of side conditions, a finite set $\omega_1$6 of indices (ordinals $\omega_1$7), finite set $\omega_1$8 of "coordinates", finite partial characteristic functions $\omega_1$9, and other bookkeeping data (notably, the sets $\omega_1$0 giving control over finite anti-chains).
• Ordering: Extensions refine the various components as expected and preserve the side condition structure.

Critical to the construction are two features:

1. Clause (C7) (Monotonicity via Side Condition Connectivity): Requires, for $\omega_1$1 (a combinatorially defined order on the indices using the side condition structure at each $\omega_1$2), that $\omega_1$3. This encodes the monotonicity of the chain modulo finite and ensures amalgamation compatibility.
2. Clause (C8) (Finite Differences): Ensures that the support for differences between the characteristic functions is finite, which implies that in the generic filter, the resulting sets form a chain modulo finite.

Main Results and Proof Strategy

The forcing $\omega_1$4 is shown to have the following properties:

• Properness: $\omega_1$5 is proper and strongly $\omega_1$6-proper relative to the collection of “small” and “large” side condition models, guaranteeing the preservation of $\omega_1$7 and higher cardinals in the extension.
• $\omega_1$8-Knaster: Under $\omega_1$9, the poset is $^{\omega_1}\omega_1$0-Knaster, so all cardinals are preserved.
• Generic Chain: In the generic extension, a strong chain of length $^{\omega_1}\omega_1$1 in $^{\omega_1}\omega_1$2 is constructed from the union of the partial characteristic functions in the filter.

The argument involves intricate amalgamation lemmas for symmetric systems, extending key commutativity results for intersection and isomorphism operations, and a highly technical control over chain-connectivity at the finite level (to manage the failure cases blocking naive fusion).

A pivotal technical novelty is that the use of symmetric systems of two types is sufficient, despite prior suggestions that side conditions of three types (e.g., with models of size $^{\omega_1}\omega_1$3) might be required. The construction tightly organizes the interaction of small and large models in the side conditions to provide the desired combinatorial and preservation features without overshooting and introducing unwanted collapse.

Implications and Comparisons

• The paper extends the reach of forcing constructions for strong chains in $^{\omega_1}\omega_1$4 by one cardinal beyond previous techniques. Earlier, Koszmider obtained strong $^{\omega_1}\omega_1$5-chains (under the assumption of $^{\omega_1}\omega_1$6 and via c.c.c. proper forcing with morasses), and the current work raises the bound to $^{\omega_1}\omega_1$7 with tools that remain tractable and modular.
• The result partially answers, for the case of subsets (as opposed to function chains in $^{\omega_1}\omega_1$8), the general question about the possible lengths of strong chains mod finite in the uncountable context, showing that the gap between lower and upper bounds is not as tight as previously thought, especially in the absence of large cardinal or chang's conjecture obstructions.
• The paper demonstrates that side condition forcings with only two types of models, when arranged in symmetric systems, suffice to push large combinatorial constructions further than was previously assumed possible without moving to systems of higher complexity. This insight may influence future constructions involving higher analogues of combinatorial structures, properness, and even specialized forms of forcing axioms.
• The authors conjecture that the construction, with mild modifications, could yield a consistency result for strong chains of functions of length $^{\omega_1}\omega_1$9 in $\omega_2$0, further closing the gap compared to the limitations established by Shelah and Inamdar.

Conclusion

This paper achieves a consistency result for the existence of a strong chain of length $\omega_2$1 in $\omega_2$2 increasing modulo finite, via forcing with symmetric systems of models of two types as side conditions. The construction synthesizes elements from prior work by Koszmider, Veličković–Venturi, and recent symmetric system techniques, and supplies a new upper bound for strong chains in the generalized Baire space setting, supporting future advances both in pure set-theoretic combinatorics and the analysis of higher cardinal invariants.

Main Theorem: If GCH holds, there is a cardinal-preserving proper forcing (with the $\omega_2$3-Knaster property) adding a strong $\omega_2$4-chain in $\omega_2$5 increasing modulo finite (2604.15894).