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A complete Lω1ω-sentence characterizing ℵ1

Published online by Cambridge University Press:  12 March 2014

Julia F. Knight
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Here an example will be given of a complete Lω1ω-sentence with a model of power ℵ1 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 Lω1ω-sentence is said to be complete if its models all satisfy the same Lω1ω-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 Lω1ω-sentence. Such a sentence is called a Scott sentence for the structure. An uncountable L-structure need not satisfy any complete Lω1ω -sentence.

A complete Lω1ω-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 Lω1ω -sentences will be denoted by CC. By a result of Lopez-Escobar [3], if k ∈ CC, k <⊐ω1.

Assuming GCH (so that ⊐α = ℵα, ), Malitz [4] showed that CC = {ℵα: α < ω1}.

Without assuming GCH, Baumgartner [1] showed that ⊐α ∈ CC for all α < ω1.

Without GCH, it is unknown whether ℵn ∈ CC for n ≥ 2. Now it will be shown that ℵ1, ∈ CC.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 1 , March 1977 , pp. 59 - 62
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

[1]Baumgartner, J., The Hanf number for complete Lω1ω-sentences (without GCH), this Journal, vol. 39 (1974), pp. 575–578.Google Scholar
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