Modeloids. I

Benda, Miroslav

doi:10.1090/S0002-9947-1979-0530044-4

Modeloids. I
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by Miroslav Benda PDF

Trans. Amer. Math. Soc. 250 (1979), 47-90 Request permission

Abstract:

If A is a set and Â is the collection of finite nonrepeating sequences of its elements then a modeloid E on A is an equivalence relation on Â which preserves length, is hereditary, and is invariant under the action of permutations. The pivotal operation on modeloids is the derivative. The theory of this operation turns out to be very rich with connections leading to diverse branches of mathematics. For example, in §3 we associate an action space with a modeloid and in §5 we characterize the action spaces which are associated with the basic modeloids, i.e., those which are derivatives of themselves. What emerges is a kind of stability for the action space. We then show that action spaces with this stability can be approximated by finite actions and, subject to certain requirements, this approximation is unique (see Proposition 5.7). Algebraically, the countable basic modeloids correspond to closed subgroups of the symmetric groups. This and the study of automorphisms of modeloids let us show, without any algebra, that the only nontrivial normal subgroups of the finite ($\left ( { \geqslant 5} \right )$) symmetric groups are the alternating groups. The last section gives, hopefully, credence to the thesis that the essence of model theory is the study of modeloids.

References

M. Benda, Reduced products and nonstandard logics, J. Symbolic Logic 34 (1969), 424–436. MR 250860, DOI 10.2307/2270907
Miroslav Benda, Construction of models from groups of permutations, J. Symbolic Logic 40 (1975), no. 3, 383–388. MR 384530, DOI 10.2307/2272160

Model theory

A. Ehrenfeucht, An application of games to the completeness problem for formalized theories, Fund. Math. 49 (1960/61), 129–141. MR 126370, DOI 10.4064/fm-49-2-129-141
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
Roland Fraïssé, Sur quelques classifications des systèmes de relations, Publ. Sci. Univ. Alger. Sér. A 1 (1954), 35–182 (1955) (French). MR 69236
Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR 0074810, DOI 10.1090/coll/036
Carol R. Karp, Finite-quantifier equivalence, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 407–412. MR 0209132
Jon Barwise (ed.), The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, No. 72, Springer-Verlag, Berlin-New York, 1968. MR 0234827
C. Ryll-Nardzewski, On Borel measurabilty of orbits, Fund. Math. 56 (1964), 129–130. MR 172810, DOI 10.4064/fm-56-1-129-130
C. Ryll-Nardzewski, On the categoricity in power $\leq \aleph _{0}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7 (1959), 545–548. (unbound insert) (English, with Russian summary). MR 0111681
Gerald E. Sacks, Saturated model theory, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1972. MR 0398817