set theory

set theory : 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 conce....   More from Wikipedia

set theory : In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF..   More from Wikipedia

 

  Carbon Based Lifeforms - Set Theory

Question : Set theory seems to be based on common sense. The simple fact related to the theory is that people can classify balls according to their colors , put the balls with the same color into one bag or perform other similar operation. But the set theory has not been tested rigorously in experimental science. Is set theory good enough to be the foundation of mathematics? What is the alternative to set theory, which can serve as the foundation of mathematics?

Answer : Mathematics does not require to be "tested rigorously in experimental science." It has nothing to do with that. Set theory is founded on the logical axioms of the Zermelo Frankel axiom system. The statements of this system (translated / reduced / simplified into common language) are: 1. Two sets are equal if they contain exactly the same elements. 2. No set is an element of itself. 3. If P(x) is a statement about x, and A is a set, then { x in A : P(x) is true } is a set. 4. If X and Y are sets, then there is a set Z containing both of them, e.g. Z = {X,Y}. 5. If F is a family of sets, then there is a set A containing every element of any set in F. (This defines the union of sets.) 6. The domain and image of any function is a set. 7. There is at least one infinite set. 8. For any set S there is a set P such that P contains exactly the subsets of S. How do you think this is supposed to be tested in experimental science? None of these concepts exist in the r....   More from Yahoo Answers

Question : I get the idea of set theory however I do not understand the axiom of choice nor why it is significant.

Answer : The axiom of choice says that the Cartesian product of nonempty sets is nonempty. That is (for example), if you have sets A_1, A_2, A_3, A_4, ..., and each set contains at least one element, then the set A_1 x A_2 x A_3 x ... contains at least one element. Actually, that's not quite what the axiom of choice says, but what I've given is an equivalent statement that's perhaps slightly easier to understand. The axiom of choice is not a theorem; it is an assertion that is (usually) part of the definition of what is meant by the word "set." --- Within set theory, one has a list of axioms which are ways in which we think that sets are supposed to behave. One of the most popular set theories is Zermelo-Fraenkel Set Theory (which I will call ZF from now on). Some examples of axioms in this theory are * If two sets have the same elements, then they are equal. * If a and b are two sets, then there exists a set whose elements are a and b. * There exists an infinite set. Whe....   More from Yahoo Answers