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Representing sets of ordinals as countable unions of sets in the core model
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Titel: |
Representing sets of ordinals as countable unions of sets in the core model |
In: | Transactions of the American Mathematical Society, 317, 1990, 1, S. 91-126 |
veröffentlicht: |
American Mathematical Society (AMS)
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Umfang: | 91-126 |
ISSN: |
0002-9947 1088-6850 |
DOI: | 10.1090/s0002-9947-1990-0939805-5 |
Zusammenfassung: | <p>We prove the following theorems. <bold>Theorem 1</bold> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal not-sign 0 Superscript number-sign Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">¬<!-- ¬ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>0</mml:mn> <mml:mi mathvariant="normal">#<!-- # --></mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\neg {0^\# })</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <italic>Every set of ordinals which is closed under primitive recursive set functions is a countable union of sets in</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <bold>Theorem 2.</bold> (No inner model with an Erdàs cardinal, i.e. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa right-arrow left-parenthesis omega 1 right-parenthesis Superscript greater-than omega"> <mml:semantics> <mml:mrow> <mml:mi>κ<!-- κ --></mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>></mml:mo> <mml:mi>ω<!-- ω --></mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\kappa \to {({\omega _1})^{ > \omega }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.) <italic>For every ordinal</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>there is in</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>an algebra on</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>with countably many operations such that every subset of</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>closed under the operations of the algebra is a countable union of sets in</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> |
Format: | E-Article |
Quelle: | American Mathematical Society (AMS) (CrossRef) |
Sprache: | Englisch |