Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T21:50:06.542Z Has data issue: false hasContentIssue false
  • English
  • Français

A complete Lω1ω-sentence characterizing ℵ1

Published online by Cambridge University Press:  12 March 2014

Julia F. Knight
Get access Rights & Permissions [Opens in a new window]

Extract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baumgartner, J., The Hanf number for complete Lω1ω-sentences (without GCH), this Journal, vol. 39 (1974), pp. 575–578.Google Scholar
[2]Knight, J., Generic expansions of structures, this Journal, vol. 38 (1973), pp. 561–570.Google Scholar
[3]Lopez-Escobar, E. G. K., On defining well-orderings, Fundamenta Mathematicae, vol. 59 (1966), pp. 13–21.CrossRefGoogle Scholar
[4]Malitz, J., The Hanf number for complete Lω1ω-sentences, The syntax and semantics of infinitary languages (Barwise, J., Editor), Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 166–181.CrossRefGoogle Scholar
[5]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers, The theory of models (Proceedings of the 1963 International Symposium at Berkeley) (Addison, , Henkin, , and Tarski, , Editors), North-Holland, Amsterdam, 1965, pp. 329–341.Google Scholar
[6]Shelah, S., Categoricity in ℵ1 of sentences in Lω1ω, Israel Journal of Mathematics, vol. 20 (1975), pp. 127–148.CrossRefGoogle Scholar