On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic
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&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/726 |
| _version_ | 1797623838941380608 |
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| author | Vladimir Kanovei Vassily Lyubetsky |
| author_facet | Vladimir Kanovei Vassily Lyubetsky |
| author_sort | Vladimir Kanovei |
| collection | DOAJ |
| description | 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>. |
| first_indexed | 2024-03-11T09:34:30Z |
| format | Article |
| id | doaj.art-7fd045fa6d7944ec942f06d901cec70c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:34:30Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7fd045fa6d7944ec942f06d901cec70c2023-11-16T17:23:42ZengMDPI AGMathematics2227-73902023-02-0111372610.3390/math11030726On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano ArithmeticVladimir Kanovei0Vassily Lyubetsky1Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, RussiaInstitute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, RussiaWe 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>.https://www.mdpi.com/2227-7390/11/3/726forcingprojective well-orderingsprojective classesJensen’s forcing |
| spellingShingle | Vladimir Kanovei Vassily Lyubetsky On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic Mathematics forcing projective well-orderings projective classes Jensen’s forcing |
| title | On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic |
| title_full | On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic |
| title_fullStr | On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic |
| title_full_unstemmed | On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic |
| title_short | On the Significance of Parameters in the Choice and Collection Schemata in the 2nd Order Peano Arithmetic |
| title_sort | on the significance of parameters in the choice and collection schemata in the 2nd order peano arithmetic |
| topic | forcing projective well-orderings projective classes Jensen’s forcing |
| url | https://www.mdpi.com/2227-7390/11/3/726 |
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