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- Hypotéza singulárních kardinálů (někdy také označovaná zkratkou SCH) je tvrzení z oboru teorie množin, které (pokud je přijato) zjednodušuje výpočet kardinální mocniny. Toto tvrzení bylo formulováno v roce 1974 v následujícím tvaru: (cs)
- Die singuläre-Kardinalzahlen-Hypothese, nach der englischen Bezeichnung singular cardinals hypothesis auch als SCH abgekürzt, ist eine von den üblichen Axiomen der Mengenlehre unabhängige Aussage, die daher weder bewiesen noch widerlegt werden kann. Sie taucht im Rahmen der Untersuchungen über die Kontinuumshypothese auf. (de)
- In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal. According to Mitchell (1992), the singular cardinals hypothesis is: If κ is any singular strong limit cardinal, then 2κ = κ+. Here, κ+ denotes the successor cardinal of κ. Since SCH is a consequence of GCH, which is known to be consistent with ZFC, SCH is consistent with ZFC. The negation of SCH has also been shown to be consistent with ZFC, if one assumes the existence of a sufficiently large cardinal number. In fact, by results of Moti Gitik, ZFC + the negation of SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of Mitchell order κ++. Another form of the SCH is the following statement: 2cf(κ) < κ implies κcf(κ) = κ+, where cf denotes the cofinality function. Note that κcf(κ)= 2κ for all singular strong limit cardinals κ. The second formulation of SCH is strictly stronger than the first version, since the first one only mentions strong limits. From a model in which the first version of SCH fails at ℵω and GCH holds above ℵω+2, we can construct a model in which the first version of SCH holds but the second version of SCH fails, by adding ℵω Cohen subsets to ℵn for some n. Jack Silver proved that if κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Silver's original proof used . The following important fact follows from Silver's theorem: if the singular cardinals hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals. In particular, then, if is the least counterexample to the singular cardinals hypothesis, then . The negation of the singular cardinals hypothesis is intimately related to violating the GCH at a measurable cardinal. A well-known result of Dana Scott is that if the GCH holds below a measurable cardinal on a set of measure one—i.e., there is normal -complete ultrafilter D on such that , then . Starting with a supercompact cardinal, Silver was able to produce a model of set theory in which is measurable and in which . Then, by applying Prikry forcing to the measurable , one gets a model of set theory in which is a strong limit cardinal of countable cofinality and in which —a violation of the SCH. Gitik, building on work of Woodin, was able to replace the supercompact in Silver's proof with measurable of Mitchell order . That established an upper bound for the consistency strength of the failure of the SCH. Gitik again, using results of inner model theory, was able to show that a measurable cardinal of Mitchell order is also the lower bound for the consistency strength of the failure of SCH. A wide variety of propositions imply SCH. As was noted above, GCH implies SCH. On the other hand, the proper forcing axiom, which implies and hence is incompatible with GCH also implies SCH. Solovay showed that large cardinals almost imply SCH—in particular, if is strongly compact cardinal, then the SCH holds above . On the other hand, the non-existence of (inner models for) various large cardinals (below a measurable cardinal of Mitchell order ) also imply SCH. (en)
- 집합론에서 특이 기수 가설(特異基數假說, 영어: singular cardinals hypothesis, 약자 SCH)은 기수의 거듭제곱이 연속체 함수 로부터 완전히 결정된다는 명제이다. 통상적인 집합론 공리계(선택 공리를 추가한 체르멜로-프렝켈 집합론)와 독립적이다. (ko)
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- Hypotéza singulárních kardinálů (někdy také označovaná zkratkou SCH) je tvrzení z oboru teorie množin, které (pokud je přijato) zjednodušuje výpočet kardinální mocniny. Toto tvrzení bylo formulováno v roce 1974 v následujícím tvaru: (cs)
- Die singuläre-Kardinalzahlen-Hypothese, nach der englischen Bezeichnung singular cardinals hypothesis auch als SCH abgekürzt, ist eine von den üblichen Axiomen der Mengenlehre unabhängige Aussage, die daher weder bewiesen noch widerlegt werden kann. Sie taucht im Rahmen der Untersuchungen über die Kontinuumshypothese auf. (de)
- 집합론에서 특이 기수 가설(特異基數假說, 영어: singular cardinals hypothesis, 약자 SCH)은 기수의 거듭제곱이 연속체 함수 로부터 완전히 결정된다는 명제이다. 통상적인 집합론 공리계(선택 공리를 추가한 체르멜로-프렝켈 집합론)와 독립적이다. (ko)
- In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal. According to Mitchell (1992), the singular cardinals hypothesis is: If κ is any singular strong limit cardinal, then 2κ = κ+. Here, κ+ denotes the successor cardinal of κ. Another form of the SCH is the following statement: 2cf(κ) < κ implies κcf(κ) = κ+, (en)
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