Information AboutSet Theory |
| CATEGORIES ABOUT SET THEORY | |
| set theory | |
| formal methods | |
| mathematical logicset theory | |
| formal methods | |
| mathematical logic | |
| mathematics | |
| theories | |
| mathematical logic | |
|
In Naive Set Theory , sets are introduced and understood using what is taken to be the self-evident concept of sets as collections of objects considered as a whole. In Axiomatic Set Theory , the concepts of sets and set membership are defined indirectly by first postulating certain Axiom s which specify their properties. In this conception, sets and set membership are fundamental concepts like Point and Line in Euclidean Geometry , and are not themselves directly defined. OBJECTIONS TO SET THEORY Since its inception, there have been some mathematicians who have Objected To Using Set Theory As A Foundation For Mathematics , claiming that it is just a game which includes elements of fantasy. Errett Bishop dismissed set theory as "God's mathematics, which we should leave for God to do." Also Ludwig Wittgenstein questioned especially the handling of infinities, which concerns also ZF . Wittgenstein's views about foundations of mathematics have been criticised by Paul Bernays , and closely investigated by Crispin Wright , among others. The most frequent objection to set theory is the Constructivist view that mathematics is loosely related to computation and that Naive Set Theory is being formalised with the addition of noncomputational elements. Topos Theory has been proposed as an alternative to traditional axiomatic set theory. Topos theory can be used to interpret various alternatives to set theory such as Constructivism , Fuzzy Set Theory , finite set theory, and Computable set theory. CULTURAL REFERENCES
SEE ALSO
|
|
|