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ResExt021703

# ResExt021703 - 1 RESTRICTIONS AND EXTENSIONS by Harvey M...

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1 RESTRICTIONS AND EXTENSIONS by Harvey M. Friedman Ohio State University Princeton University [email protected] http://www.math.ohio-state.edu/~friedman/ February 17, 2003 Abstract. We consider a number of statements involving restrictions and extensions of algebras, and derive connections with large cardinal axioms. 1. Introduction. By an algebra M we mean a nonempty set dom(M) together with a finite list of functions on dom(M) of various finite arities 0. The signature of M is the list of arities of the functions of M. Let M,N be algebras. We say that M is a restriction of N if and only if M,N have the same signature, dom(M) dom(N), and the functions of M are restrictions of the respective functions of N. We say that N is an extension of M if and only if M is a restriction of N. We use k , l for cardinals throughout the paper. We write R( k ,fg) if and only if every algebra of cardinality k has a proper restriction with the same finitely generated restrictions up to isomorphism. We write R( k , l ) if and only if every algebra of cardinality k has a proper restriction with the same restrictions of cardinality l up to isomorphism. We write E( k ,fg) if and only if every algebra of cardinality k has a proper extension with the same finitely generated restrictions up to isomorphism. We write E( k , l ) if and only if every algebra of cardinality k has a proper extension with the same restrictions of cardinality l up to isomorphism. In this paper, we will restrict attention to the cases l = w . All theorems are proved in ZFC.

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2 R( k ,fg), R( k , w ) are easily treated as follows. THEOREM 1.1. The following are equivalent. i) R( k ,fg); ii) R( k , w ); iii) k > 2 w . Our main results concern sufficiently large k . THEOREM 1.2. The following are equivalent. ii) for all sufficiently large k , E( k ,fg); ii) for all sufficiently large k , E( k , w ); iii) there exists a measurable cardinal.
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