What Is The Universal Mapping Property As Used In Set Theory, And What Are Some Examples Of It?

[culley96]

set theory mathematics, The universal mapping property. What are some examples of this property?

[Bernd]

We say that h:X® UA is a universal map from the object... DEFINITION 7.1.1 We say that h:X® UA is a universal map from the object X Î obS to the functor  U:A® S if, whenever f:X® UQ is another morphism with Q Î obA, there is a unique A-map p:A® Q such that h;U p = f(cf Remark 3.6.3 for posets).   The Essence of Set Theory 2. Origins of Set Theory 3. The Continuum Hypothesis 4. Axiomatic Set Theory 5. The Axiom of Choice 6. Inner Models 7. Independence Proofs 8. Large Cardinals 9. Descriptive Set Theory Bibliography Other Internet Resources Related Entries 1. The Essence of Set Theory The objects of study of Set Theory are sets. As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms. The language of set theory is based on a single fundamental relation, called membership. We say that A is a member of B(in symbols A ? B), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. In practice, one considers sets of numbers, sets of points, sets of functions, sets of some other sets and so on. In theory, it is not necessary to distinguish between objects that are members and objects that contain members -- the only objects one needs for the theory are sets. See the supplement for further discussion. Using the basic construction principles, and assuming the existence of infinite sets, one can define numbers, including integers, real numbers and complex numbers, as well as functions, functionals, geometric and topological concepts, and all objects studied in mathematics. In this sense, set theory serves as Foundations of Mathematics. The significance of this is that all questions of provability(or unprovability) of mathematical statements can be in principle reduced to formal questions of formal derivability from the generally accepted axioms of Set Theory. While the fact that all of mathematics can be reduced to a formal system of set theory is significant, it would hardly be a justification for the study of set theory. It is the internal structure of the theory that makes it worthwhile, and it turns out that this internal structure is enormously complex and interesting. Moreover, the study of this structure leads to significant questions about the nature of the mathematical universe. The fundamental concept in the theory of infinite sets is the cardinality of a set. Two sets A and B have the same cardinality if there exists a mapping from the set A onto the set B which is one-to-one, that is, it assigns each element of A exactly one element of B. It is clear that when two sets are finite, then they have the same cardinality if and only if they have the same number of elements. One can extend the concept of the ?number of elements? to arbitrary, even infinite, sets. It is not apparent at first that there might be infinite sets of different cardinalities, but once this becomes clear, it follows quickly that the structure so described is rich indeed. 2. Origins of Set Theory The birth of Set Theory dates to 1873 when Georg Cantor proved the uncountability of the real line.(One could even argue that the exact birthdate is December 7, 1873, the date of Cantor's letter to Dedekind informing him of his discovery.) Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for ?actual infinity.? The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets, and infinity appears only as ?a manner of speaking?, to paraphrase Friedrich Gauss. The fact that the set of all positive integers has a proper subset, like the set of squares {1, 4, 9, 16, 25,...} of the same cardinality(using modern terminology) was considered somewhat paradoxical(this had been discussed at length by Galileo among others). Such apparent paradoxes prevented Bernhard Bolzano in 1840s from developing set theory, even though some of his ideas are precursors of Cantor's work.(It should be mentioned that Bolzano, an accomplished mathematician himself, coined the word Menge(= set) that Cantor used for objects of his theory.) Sources: http://plato.stanford.edu/entries/set-theory/#2 BarefootInformation 86 months ago Please sign in to give a compliment. Please verify your account to give a compliment. Please sign in to send a message. Please verify your account to send a message.

[Chuioke]

DEFINITION 7.1.1 We say that h:X® UA is a universal map from the object X Î obS to the functor  U:A® S if, whenever f:X® UQ is another morphism with Q Î obA, there is a unique A-map p:A® Q such that h;U p = f(cf Remark 3.6.3 for posets).   The Essence of Set Theory 2. Origins of Set Theory 3. The Continuum Hypothesis 4. Axiomatic Set Theory 5. The Axiom of Choice 6. Inner Models 7. Independence Proofs 8. Large Cardinals 9. Descriptive Set Theory Bibliography Other Internet Resources Related Entries 1. The Essence of Set Theory The objects of study of Set Theory are sets. As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms. The language of set theory is based on a single fundamental relation, called membership. We say that A is a member of B(in symbols A ? B), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. In practice, one considers sets of numbers, sets of points, sets of functions, sets of some other sets and so on. In theory, it is not necessary to distinguish between objects that are members and objects that contain members -- the only objects one needs for the theory are sets. See the supplement for further discussion. Using the basic construction principles, and assuming the existence of infinite sets, one can define numbers, including integers, real numbers and complex numbers, as well as functions, functionals, geometric and topological concepts, and all objects studied in mathematics. In this sense, set theory serves as Foundations of Mathematics. The significance of this is that all questions of provability(or unprovability) of mathematical statements can be in principle reduced to formal questions of formal derivability from the generally accepted axioms of Set Theory. While the fact that all of mathematics can be reduced to a formal system of set theory is significant, it would hardly be a justification for the study of set theory. It is the internal structure of the theory that makes it worthwhile, and it turns out that this internal structure is enormously complex and interesting. Moreover, the study of this structure leads to significant questions about the nature of the mathematical universe. The fundamental concept in the theory of infinite sets is the cardinality of a set. Two sets A and B have the same cardinality if there exists a mapping from the set A onto the set B which is one-to-one, that is, it assigns each element of A exactly one element of B. It is clear that when two sets are finite, then they have the same cardinality if and only if they have the same number of elements. One can extend the concept of the ?number of elements? to arbitrary, even infinite, sets. It is not apparent at first that there might be infinite sets of different cardinalities, but once this becomes clear, it follows quickly that the structure so described is rich indeed. 2. Origins of Set Theory The birth of Set Theory dates to 1873 when Georg Cantor proved the uncountability of the real line.(One could even argue that the exact birthdate is December 7, 1873, the date of Cantor's letter to Dedekind informing him of his discovery.) Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for ?actual infinity.? The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets, and infinity appears only as ?a manner of speaking?, to paraphrase Friedrich Gauss. The fact that the set of all positive integers has a proper subset, like the set of squares {1, 4, 9, 16, 25,...} of the same cardinality(using modern terminology) was considered somewhat paradoxical(this had been discussed at length by Galileo among others). Such apparent paradoxes prevented Bernhard Bolzano in 1840s from developing set theory, even though some of his ideas are precursors of Cantor's work.(It should be mentioned that Bolzano, an accomplished mathematician himself, coined the word Menge(= set) that Cantor used for objects of his theory.)