Article
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On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic
Version 1
: Received: 13 December 2022 / Approved: 14 December 2022 / Online: 14 December 2022 (09:58:35 CET)
Version 2 : Received: 23 December 2022 / Approved: 26 December 2022 / Online: 26 December 2022 (12:26:30 CET)
Version 2 : Received: 23 December 2022 / Approved: 26 December 2022 / Online: 26 December 2022 (12:26:30 CET)
A peer-reviewed article of this Preprint also exists.
Kanovei, V.; Lyubetsky, V. On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic. Mathematics 2023, 11, 726. Kanovei, V.; Lyubetsky, V. On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic. Mathematics 2023, 11, 726.
Abstract
We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either 1) the parameter-free countable axiom of choice AC* fails, or 2) AC* holds but the full countable axiom of choice AC fails in the domain of reals. In another generic extension of L, we define a set X⊆P(ω), which is a model of the parameter-free part PA2* of the 2nd order Peano arithmetic PA2, in which CA(Σ21) (Comprehension for Σ21 formulas with parameters) holds, yet an instance of Comprehension CA for a more complex formula fails. Treating the iterated Sacks forcing as a class forcing over $L_{\omega_1}$, we infer the following consistency results as corollaries. If the 2nd order Peano arithmetic PA2 is formally consistent then so are the theories: 1) PA2 + negation of AC*, 2) PA2 + AC* + negation of AC, 3) PA2* + CA(Σ21) + negation of CA.
Keywords
forcing; second-order arithmetic; Comprehension; Choice; Sacks forcing
Subject
Computer Science and Mathematics, Logic
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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