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- W2575297946 abstract "Here an example will be given of a complete L ω1ω -sentence with a model of power ℵ 1 but with no model of higher power. The continuum hypothesis is not assumed. The question of whether such an example exists was brought to the author's attention by Professor M. Makkai. An L ω1ω -sentence is said to be complete if its models all satisfy the same L ω1ω -sentences, or, equivalently, if all of the countable L -structures satisfying the sentence are isomorphic. Scott [5] showed that any countable L -structure (where L is countable) must satisfy a complete L ω1ω -sentence. Such a sentence is called a Scott sentence for the structure. An uncountable L -structure need not satisfy any complete L ω1ω -sentence. A complete L ω1ω -sentence σ is said to characterize the infinite cardinal k if σ has a model of power k but not of any higher power. The set of cardinals characterized by complete L ω1ω -sentences will be denoted by CC. By a result of Lopez-Escobar [3], if k ∈ CC, k <⊐ ω1 . Assuming GCH (so that ⊐ α = ℵ α , ), Malitz [4] showed that CC = {ℵ α : α < ω 1 }. Without assuming GCH, Baumgartner [1] showed that ⊐ α ∈ CC for all α < ω 1 . Without GCH, it is unknown whether ℵ n ∈ CC for n ≥ 2. Now it will be shown that ℵ 1 , ∈ CC." @default.
- W2575297946 created "2017-01-26" @default.
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- W2575297946 date "1977-03-01" @default.
- W2575297946 modified "2023-10-06" @default.
- W2575297946 title "A complete <i>L</i><sub>ω1ω</sub>-sentence characterizing ℵ<sub>1</sub>" @default.
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- W2575297946 doi "https://doi.org/10.2307/2272318" @default.
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