On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom
Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface <inline-formula><math xmlns="http://www.w3....
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doaj-ca03603e77204eaea8ffc4902a080e192021-07-23T13:52:33ZengMDPI AGMathematics2227-73902021-07-0191670167010.3390/math9141670On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski AxiomAli Enayat0Vladimir Kanovei1Vassily Lyubetsky2Department of Philosophy, Linguistics, and Theory of Science, University of Gothenburg, 405 30 Gothenburg, SwedenInstitute 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, RussiaExamples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Π</mi><mn>2</mn><mn>1</mn></msubsup></semantics></math></inline-formula> equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any <i>n</i>, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Π</mi><mi>n</mi><mn>1</mn></msubsup></semantics></math></inline-formula>, does not imply the existence of such a pair with the associated relation in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Σ</mi><mi>n</mi><mn>1</mn></msubsup></semantics></math></inline-formula> or in a lower class.https://www.mdpi.com/2227-7390/9/14/1670indiscernible setsLeibniz-Mycielski axiomprojective hierarchygeneric modelsordinal definabilityMiller forcing |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Ali Enayat Vladimir Kanovei Vassily Lyubetsky |
spellingShingle |
Ali Enayat Vladimir Kanovei Vassily Lyubetsky On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom Mathematics indiscernible sets Leibniz-Mycielski axiom projective hierarchy generic models ordinal definability Miller forcing |
author_facet |
Ali Enayat Vladimir Kanovei Vassily Lyubetsky |
author_sort |
Ali Enayat |
title |
On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom |
title_short |
On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom |
title_full |
On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom |
title_fullStr |
On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom |
title_full_unstemmed |
On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom |
title_sort |
on effectively indiscernible projective sets and the leibniz-mycielski axiom |
publisher |
MDPI AG |
series |
Mathematics |
issn |
2227-7390 |
publishDate |
2021-07-01 |
description |
Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Π</mi><mn>2</mn><mn>1</mn></msubsup></semantics></math></inline-formula> equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any <i>n</i>, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Π</mi><mi>n</mi><mn>1</mn></msubsup></semantics></math></inline-formula>, does not imply the existence of such a pair with the associated relation in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>Σ</mi><mi>n</mi><mn>1</mn></msubsup></semantics></math></inline-formula> or in a lower class. |
topic |
indiscernible sets Leibniz-Mycielski axiom projective hierarchy generic models ordinal definability Miller forcing |
url |
https://www.mdpi.com/2227-7390/9/14/1670 |
work_keys_str_mv |
AT alienayat oneffectivelyindiscernibleprojectivesetsandtheleibnizmycielskiaxiom AT vladimirkanovei oneffectivelyindiscernibleprojectivesetsandtheleibnizmycielskiaxiom AT vassilylyubetsky oneffectivelyindiscernibleprojectivesetsandtheleibnizmycielskiaxiom |
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