Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
Set (mathematics)A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics.
Intersection (set theory)In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to Intersection is written using the symbol "" between the terms; that is, in infix notation. For example: The intersection of more than two sets (generalized intersection) can be written as: which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
ReproducibilityReproducibility, closely related to replicability and repeatability, is a major principle underpinning the scientific method. For the findings of a study to be reproducible means that results obtained by an experiment or an observational study or in a statistical analysis of a data set should be achieved again with a high degree of reliability when the study is replicated. There are different kinds of replication but typically replication studies involve different researchers using the same methodology.
Reproducibility ProjectThe Reproducibility Project: Psychology was a crowdsourced collaboration of 270 contributing authors to repeat 100 published experimental and correlational psychological studies. This project was led by the Center for Open Science and its co-founder, Brian Nosek, who started the project in November 2011. The results of this collaboration were published in August 2015. Reproducibility is the ability to produce the same findings, using the same methodologies as the original work, but on a different dataset (for instance, collected from a different set of participants).
Replication crisisThe replication crisis (also called the replicability crisis and the reproducibility crisis) is an ongoing methodological crisis in which the results of many scientific studies are difficult or impossible to reproduce. Because the reproducibility of empirical results is an essential part of the scientific method, such failures undermine the credibility of theories building on them and potentially call into question substantial parts of scientific knowledge.
