Permissively structured transfinite electrical networks

Abstract

Transfinite electrical networks of ranks larger than 1 have previously been defined by arbitrarily joining together various infinite extremities through transfinite nodes that are independent of the networks' resistance values. Thus, some or all of those transfinite nodes may remain ineffective in transmitting current “through infinity.” In this paper, transfinite nodes are defined in terms of the paths that permit currents to ”reach infinity.” This is accomplished by defining a suitable metricd ^v on the node setN _S ^v of eachv-sectionS ^v, av-section being a maximal subnetwork whose nodes are connected by two-ended paths of ranks no larger thanv. Upon taking the completion ofN _S ^v under that metricd ^v, we identify those extremities (now calledv-terminals) that are accessible to current flows. These are used to define transfinite nodes that combine such extremities. The construction is recursive and is carried out through all the natural number ranks, and then through the first arrow rank ω and the first limit-ordinal rank ω. The recursion can be carried still further. All this provides a more natural development of transfinite networks and indeed simplifies the theory of electrical behavior for such networks.