Abstract
The topos theory gives tools for unified proofs of theorems for model theory for various semantics and logics. We introduce the notion of power and the notion of generalized quantifier in topos and we formulate sufficient condition for such quantifiers in order that they fulfil downward Skolem-Löwenheim theorem when added to the language. In the next paper, in print, we will show that this sufficient condition is fulfilled in a vast class of Grothendieck toposes for the general and the existential quantifiers.
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References
M. C. Fitting,Intuitionistic Logic, Model Theory and Forcing, North-Holland 1969.
M. P. Fourman, C. J. Mulvey andD. S. Scott,Applications of Sheaf Theory to Algebra, Analysis and Topology, Lecture Notes in Mathematics, Springer-Verlag 1979.
D. Higgs,A category approach to the Boolean-valued set theory, preprint.
P. T. Johnstone,Topos Theory, Academic Press 1977.
P. Lindström,First order predicate logic with generalized quantifiers, Theoria, Vol. 32 (1966) pp. 186–195.
G. Loullis,Sheaves and Boolean-valued model theory The Journal of Symbolic Logic Vol. 44 (1979) pp. 153–183.
S. MacLane,Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag 1971.
M. Makkai andG. E. Reyes,First Order Categorial Logic, Lecture Notes in Mathematics, Vol. 611, 1977 Springer-Verlag.
H. Rasiowa andR. Sikorski,The Mathematics of Methamathematics, Państwowe Wydawnictwo Naukowe 1963.
G. C. Wraith,Lectures on Elementary Topoi, Model Theory and Topoi, Lectures Notes in Mathematics, Vol. 445 (1975), Springer-Verlag.
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Zawadowski, M. The Skolem-Löwenheim theorem in toposes. Stud Logica 42, 461–475 (1983). https://doi.org/10.1007/BF01371634
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DOI: https://doi.org/10.1007/BF01371634