Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.
The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.
The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects. Elementary operations such as set union and intersection can be studied in this context. More advanced concepts such as cardinality are a standard part of the undergraduate mathematics curriculum.
Set theory, formalized using first-order logic, is the most common foundational system for mathematics. Beyond its use as a foundational system, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
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Q. Please explain the difference in axiomatic set theory between Euclidean and non-Euclidean geometry.
Asked by just thinking - Mon Jul 28 09:41:41 2008 - - 2 Answers - 0 Comments
A. Axiomatic set theory has nothing to do with Euclidean geometry, nor with non-Euclidean geometry. It has to do with axiomatizing set theory. Probably you want to know how the axioms for Euclidean geometry differ from the Axioms of non-Euclidean geometry. Only one axiom is different: the parallel axiom. In Euclidean geometry it says there is one and only one line parallel to a given line through a given point not on the line. This refers to lines on a plane not in 3d-space. When this is replaced with the axiom that there are no lines parallel to the given line through the given point the resulting geometry is called elliptic non-Euclidean geometry. When it is replaces with the axiom that there are at least two lines parallel to the… [cont.]
Answered by JB - Mon Jul 28 10:09:41 2008
