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This paper demonstrates that the strength of restricted forms of the axioms of choice (ACω and DC) and comprehension (CA) in set theory and second-order arithmetic depends on the projective level, the presence or absence of parameters in definitions, and the specific axiom variant.

Abstract

This research paper investigates the interplay between restricted versions of the axioms of countable choice (ACω), dependent choice (DC), and comprehension (CA) within the frameworks of Zermelo-Fraenkel set theory (ZF) and second-order Peano arithmetic (PA2).

**Bibliographic Information:** Kanovei, V., & Lyubetsky, V. (2024). On the significance of parameters and the complexity level in the Choice and Collection axioms. *arXiv preprint arXiv:2407.20098v5*.

**Research Objective:** The paper aims to clarify the relative strength and independence of various restricted forms of ACω, DC, and CA by constructing specific generic models of ZF and PA2 where certain forms hold while others fail.

**Methodology:** The authors employ techniques from set theory and forcing, particularly generalized iterations of Jensen forcing and symmetric submodels, to construct cardinal-preserving generic extensions of the constructible universe (L) that satisfy desired combinations of axioms and their negations. They introduce the concept of "normal forcing" based on iterated perfect sets and define key properties like Fusion, Structure, n-Definability, and n-Completeness to control the properties of these models.

**Key Findings:** The paper establishes the following key results:

- For each n ≥ 1, there exist models of ZF where various combinations of Π1n-DC, Π1n+1-ACω, OD-ACω, and ACω hold or fail, demonstrating the significance of the projective level and parameters in these axioms.
- For each n ≥ 1, there exists a model of PA2 with parameter-free choice (Σ1∞-ACω) where Σ1n+1-CA holds but Σ1n+2-CA fails, showing that parameters significantly impact the strength of comprehension axioms.

**Main Conclusions:** The authors conclude that the strength of restricted forms of ACω, DC, and CA is highly sensitive to the projective level, the inclusion or exclusion of parameters, and the specific axiom variant. These findings contribute to a deeper understanding of the foundational role of choice and comprehension principles in set theory and arithmetic.

**Significance:** This research significantly advances the understanding of the axiomatic foundations of mathematics, particularly by clarifying the subtle interplay between choice and comprehension principles at various levels of definability.

**Limitations and Future Research:** The paper primarily focuses on specific levels of the projective hierarchy. Further research could explore similar questions for higher projective levels or other definability classes. Investigating the consistency strength of these results relative to weaker theories is another potential avenue for future work.

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by Vladimir Kan... at **arxiv.org** 10-10-2024

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It's highly plausible that similar results could be obtained for other set-theoretic axioms beyond choice and comprehension. The core principles at play, namely the impact of parameters, the projective level of definability, and the use of forcing techniques to control these aspects within constructed models, have broad applicability in set theory.
Here's a breakdown of how these principles could be explored in the context of other axioms:
Separation/Subset Axiom Schema: Analogous to Comprehension in second-order arithmetic, the Separation axiom schema in set theory allows for the formation of subsets. One could investigate the effect of restricting the complexity of formulas allowed in Separation, both with and without parameters. For instance, is there a level of the projective hierarchy where parameter-free Separation is consistently weaker than its parameterized counterpart?
Replacement Axiom Schema: This schema asserts that the image of a set under a definable function is again a set. Similar to Separation, one could explore the consequences of restricting the complexity of formulas defining these functions, distinguishing between parameter-free and parameterized versions.
Power Set Axiom: While the Power Set axiom doesn't directly involve formulas, one could investigate weaker versions that assert the existence of restricted power sets, for example, consisting only of sets of a certain definability level. The influence of parameters could be studied by considering definability relative to a fixed set.
Cardinal Characteristics of the Continuum: Many cardinal characteristics of the continuum, which capture various combinatorial properties of the real line, are defined in terms of selection principles that resemble weakened forms of choice. Investigating the influence of parameters and projective level in the definitions of these characteristics could yield interesting results.
The techniques used to prove the results in the paper, particularly the use of iterated forcing with symmetric submodels, could be adapted and extended to study these questions. However, each axiom presents its own unique challenges and might require novel forcing constructions and methods for analyzing the resulting models.

There's a compelling argument to be made that parameter-free versions of axioms like Comprehension, Choice, and others are indeed more "natural" or intuitively justifiable from a foundational perspective, even though they might be formally weaker. Here's why:
Conceptual Clarity: Parameter-free versions align more closely with a minimalist view of set formation. They suggest that the existence of a set should depend solely on intrinsic properties expressible without reference to arbitrary external objects (parameters). This resonates with the idea that sets should be "built up from below" in a well-defined manner.
Predicativity Concerns: Parameters can introduce a degree of impredicativity, where the definition of a set might implicitly rely on a larger universe of sets that already includes the set being defined. Parameter-free versions, by avoiding this circularity, are more aligned with predicative approaches to mathematics.
Philosophical Elegance: There's an aesthetic appeal to the simplicity and self-contained nature of parameter-free axioms. They suggest a more parsimonious and conceptually elegant foundation for mathematics.
However, the weaker consistency strength of parameter-free versions presents a trade-off:
Expressive Power: Parameters are undeniably useful in practice. They allow us to express a wider range of mathematical concepts and carry out constructions that might be impossible or highly cumbersome without them.
Common Practice: Mathematics, as it's practiced, heavily relies on parameterized definitions and constructions. Restricting ourselves to parameter-free versions would significantly limit the scope of mathematics.
Ultimately, the choice between "naturalness" and expressive power is a matter of philosophical preference and depends on the goals of one's foundational framework.

The realization that the precise level of comprehension needed for developing mathematics within PA2 is not uniquely determined by its consistency has profound philosophical implications for the foundations of mathematics:
Indeterminacy of Mathematical Practice: It suggests a degree of indeterminacy in mathematical practice. There might be multiple, equally consistent, but non-equivalent ways of developing mathematics, each corresponding to a different level of comprehension. This challenges the view that there's a single, canonical way of "doing mathematics."
The Role of Philosophical Commitments: The choice of a particular level of comprehension becomes a matter of philosophical commitment rather than a matter of logical necessity. Mathematicians might be led to adopt stronger or weaker versions of comprehension based on their philosophical views about the nature of sets, definability, or the justifiability of impredicative definitions.
Limitations of Formal Systems: It highlights the limitations of formal systems in capturing the full richness of mathematical practice. Consistency, while a necessary condition, might not be sufficient to uniquely determine a "correct" or "intended" interpretation of mathematics.
Pluralism in the Foundations of Mathematics: This indeterminacy lends support to a pluralistic view of the foundations of mathematics, where different foundational frameworks, each embodying different philosophical commitments, might coexist and provide equally valid perspectives on mathematics.
Open-Ended Nature of Mathematics: It reinforces the open-ended nature of mathematics. As we explore higher levels of complexity and seek to formalize new mathematical concepts, we might discover the need for ever stronger forms of comprehension, without ever reaching a definitive "final" system.
In essence, this realization challenges a purely formalist view of mathematics and suggests a more nuanced perspective where philosophical considerations play a crucial role in shaping our understanding of the subject.

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