Set theory, different systems of

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory. Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals). Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception. Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.

Total Internal Reflection (TIR) Lens Design for Surface Mount Techonology (SMT) LED Application

In this paper, Total Internal Reflection (TIR) lens for a micro optical LED application is proposed. The aim of this paper is to design TIR lens in collimating light emitted from a surface mount technology (SMT) LED. To satisfy current technical needs, it was designed to have both thickness and diameter less than three millimeters with eleven facets where light refracts and reflects. Overall geometry of the TIR lens is numerically calculated and designed in Matlab and the central hyperboloid is designed through Code V. Manufacturability of TIR lens has been obstructing its realistic LED application because of its limitation of size. In this study, we have considered this issue and found the optimized each facet angles in which ultraprecision diamond turning is applicable. Typical blue SMT LED was modeled in LightTools to verify the decrease of solid angle. After designing, the solid angle of a LED was reduced from 120° to 20° while loosing source energy about 52%. The solid angle varies with respect to its distance from the source. The simulation results show collimation efficiency of the designed lens.

Axioms for the set-theoretic hierarchy

The axioms for Zermelo-Fraenkel (ZF) set theory are an appealing but somewhat arbitrary-seeming assortment. A survey of the axioms does not suffice to reveal the source of their attraction. Accordingly, attempts have been made to ground ZF in principles whose appeal can be felt immediately. These attempts can be classified as follows. First, some of them propose to rest the ZF axioms directly on informal doctrine. The others propose to ground the ZF axioms in other formal axioms that can be regarded as more basic. When the latter approach is taken, ZF continues to draw on informal support, but the draft is made at a more basic level.The same research can be classified in another way, according to the informal diagnosis offered for the paradoxes of set theory. In some cases, the diagnosis is that the paradoxical sets (such as the Russell set) fail to exist only because they would have to be too large; a set that would be sufficiently small must always exist. This is the doctrine of limitation of size. In other cases, the diagnosis is that the paradoxical sets fail to exist only because they would have to lie too high in a certain hierarchy of sets; a set that would lie sufficiently low in the hierarchy must always exist. This is the doctrine of the hierarchy. The present paper will investigate the latter doctrine, with the doctrine of size making a brief appearance at the end.