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A theorem of Neumann-Lara and Urrutia [3] is generalized from the plane to arbitrary n-dimensional Euclidean space R n , solving Problem 2 of [3]. By an n-ball we mean a set of the form { Theorem 1. For each n ≥ 1 there is c n > 0 such that for any finite set X Õ R n there is A Õ X, |A| ≤ 1/ 2 (n+3) , having the following property: if B ⊇ A is an n-ball,… (More)

Finite set theory, here denoted ZF fin , is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZF fin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the Bernays-Rieger method of… (More)

We divide the class of infinite computable trees into three types. For the first and second types, 0 computes a nontrivial self-embedding while for the third type 0 computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up… (More)