| Summary: | We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe <b>L</b> in which the axioms of <b>ZF</b> hold and in addition either (1) the parameter-free countable axiom of choice <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi mathvariant="bold">AC</mi><mi>ω</mi><mo>*</mo></msubsup></semantics></math></inline-formula> fails, or (2) <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi mathvariant="bold">AC</mi><mi>ω</mi><mo>*</mo></msubsup></semantics></math></inline-formula> holds but the full countable axiom of choice <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold">AC</mi><mi>ω</mi></msub></semantics></math></inline-formula> fails in the domain of reals. In another generic extension of <b>L</b>, we define a set <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>⊆</mo><mi mathvariant="script">P</mi><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which is a model of the parameter-free part <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi mathvariant="bold">PA</mi><mn>2</mn><mo>*</mo></msubsup></semantics></math></inline-formula> of the 2nd order Peano arithmetic <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold">PA</mi><mn>2</mn></msub></semantics></math></inline-formula>, in which <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">CA</mi><mo>(</mo><msubsup><mi mathvariant="sans-serif">Σ</mi><mn>2</mn><mn>1</mn></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> (Comprehension for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi mathvariant="sans-serif">Σ</mi><mn>2</mn><mn>1</mn></msubsup></semantics></math></inline-formula> formulas with parameters) holds, yet an instance of Comprehension <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="bold">CA</mi></semantics></math></inline-formula> for a more complex formula fails. Treating the iterated Sacks forcing as a class forcing over <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold">L</mi><msub><mi>ω</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>, we infer the following consistency results as corollaries. If the 2nd order Peano arithmetic <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi mathvariant="bold">PA</mi><mn>2</mn></msub></semantics></math></inline-formula> is formally consistent then so are the theories: (1) <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">PA</mi><mn>2</mn></msub><mo>+</mo><mo>¬</mo><msubsup><mi mathvariant="bold">AC</mi><mi>ω</mi><mo>*</mo></msubsup></mrow></semantics></math></inline-formula>, (2) <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">PA</mi><mn>2</mn></msub><mo>+</mo><msubsup><mi mathvariant="bold">AC</mi><mi>ω</mi><mo>*</mo></msubsup><mo>+</mo><mo>¬</mo><msub><mi mathvariant="bold">AC</mi><mi>ω</mi></msub></mrow></semantics></math></inline-formula>, (3) <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">PA</mi><mn>2</mn><mo>*</mo></msubsup><mo>+</mo><mi mathvariant="bold">CA</mi><mrow><mo>(</mo><msubsup><mi mathvariant="sans-serif">Σ</mi><mn>2</mn><mn>1</mn></msubsup><mo>)</mo></mrow><mo>+</mo><mo>¬</mo><mi mathvariant="bold">CA</mi></mrow></semantics></math></inline-formula>.
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