China 
Africa 
Explore chapters and articles related to this topic
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
It is somewhat confusing, but this question has no answer. The problem is much more general and deals with the idea of completeness of the axiomatic number theories (and, therefore, the foundations of mathematics as well) and it is connected with the famous result of Gödel, who showed that there may be some statements in classical number theories which cannot be assigned either “true” or “false” values. These include the problem stated above.
Most-intersection of countable sets
Published in Journal of Applied Non-Classical Logics, 2021
One fundamental problem in mathematics and science is the specification of an average or majority dynamics of a given collection of systems. On the one hand, if the given collection is finite, a straightforward solution is to use statistical or approximation methods. The literature is full of different solutions each of which may be based on different parameters and criteria. On the other hand, the same problem for an infinite collection of systems is usually considered to be more sophisticated. Since set theory is the foundation of mathematics, and that all mathematical entities can be formally described in terms of sets, a basic way of describing a system is by using sets of objects. Hence, one can translate the problem of finding an average dynamics of a collection of systems to a problem of finding a set with an overall membership characteristic that interpolates and roughly fits the characteristic of the sets in the given collection. A trivial solution is to take the intersection of all sets in the collection so as to obtain a sub-characteristic that is common to all sets. But taking merely set-intersections causes too much information loss, particularly in the case of infinite collections. Imagine an infinite collection of sets , for , such that is defined as the set of first i prime numbers. Clearly, each is finite. Furthermore, each prime number p is to be found in infinitely many 's and not to be found only in finitely many 's. If we take the intersection of the collection , we get the empty set since . However, this leads to information loss when finding the majority property of the given collection. In finding the average characteristic in this case, it is reasonable to include every prime number p as a part of this interpolation process since every such p is a member of the ‘majority’ of 's.