On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic

Vladimir Kanovei () and Vassily Lyubetsky ()
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Vladimir Kanovei: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
Vassily Lyubetsky: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia

Mathematics, 2023, vol. 11, issue 3, 1-19

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 ( ? ) , which is a model of the parameter-free part PA 2 * of the 2nd order Peano arithmetic PA 2 , in which CA ( ? 2 1 ) (Comprehension for ? 2 1 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 ? 1 , we infer the following consistency results as corollaries. If the 2nd order Peano arithmetic PA 2 is formally consistent then so are the theories: (1) PA 2 + ¬ AC ? * , (2) PA 2 + AC ? * + ¬ AC ? , (3) PA 2 * + CA ( ? 2 1 ) + ¬ CA .

Keywords: forcing; projective well-orderings; projective classes; Jensen’s forcing (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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