A Hamiltonian Property of Connected Sets in the Alternative Set Theory

The representation of indiscernibility phenomena by π-equivalences and of accessibility phenomena by σ-equivalences enables a graph-theoretical formulation of topological notions in the alternative set theory. Generalizing the notion of Hamiltonian graph we will introduce the notion of Hamiltonian embedding and prove that for any finite graph without isolated vertices there is a Hamiltonian embedding into any infinite set connected with respect to some πor σ-equivalence. Roughly speaking, in some sense this means that each such an infinite connected set, (in particular, each connected set in a complete metrizable topological space), contains each finite graph inside, and even is exhausted by the images of its edges. Moreover, the main Theorem 3, dealing with the so called deeply connected sets, is in fact a theorem of nonstandard arithmetic. 1. A Brief Outline of Alternative Set Theory The alternative set theory (AST), similarly as nonstandard analysis, enables to treat the infinity phenomenon emerging on formally finite collections of objects. The basic collections of objects, considered itselves as selfstanding objects, AST deals with, are called classes. Sharp classes with definite boundaries are called sets. From the point of view of classical set theory, all the sets in AST are finite, as they are subject to the axiom scheme of induction for set-theoretical formulas. Enormously large “finite” sets, however, will be called infinite. In AST we define a class to be finite if each its subclass is a set. Every finite class obviously is a set, but not vice versa. Accepting the existence of infinite sets, we accept henceforth the existence of proper semisets, i.e., of proper classes which are subclasses of (formally finite!) sets. Thus we can see that, in contradistinction to proper classes in, e.g., Gödel-Bernays set theory, proper classes in AST need not be “larger” then sets. Proper semisets rather arise as mathematical counterparts of nonsharp collections without definite boundaries. Small characters always denote sets, capital ones are used for classes. When a capital letter denotes a set, it will explicitly be pointed out. Received April 22, 1993. 1980 Mathematics Subject Classification (1991 Revision). Primary 03E70, 05C45, 54J05; Secondary 03H05, 03H15, 05C10, 54D05.