Zeroth-order logic
First-order logic without variables or quantifiers
Vicious circle principle
Principle prohibiting the defining of objects using properties dependent on said object However, it also blocks one standard definition of the natural numbers. First, we define a property as being "hereditary" if, whenever a number n has the property, so does n +1. Then we say that x has the property of being a natural number if and only if it has every hereditary property that 0 has. This definition is blocked, because it defines "natural number" in terms of the totality of all hereditary properties, but "natural number" itself would be such a hereditary property, so the definition is circular in this sense. Most modern mathematicians and philosophers of mathematics think that this particular definition is not circular in any problematic sense, and thus they reject the vicious circle principle. This principle was the reason for Russell's development of the ramified theory of types rather than the theory of simple types. See also[edit] References[edit] External links[edit]
Structure
Arrangement of interrelated elements in an object/system, or the object/system itself Load-bearing[edit] Buildings, aircraft, skeletons, anthills, beaver dams, bridges and salt domes are all examples of load-bearing structures. The structure elements are combined in structural systems. Load-bearing biological structures such as bones, teeth, shells, and tendons derive their strength from a multilevel hierarchy of structures employing biominerals and proteins, at the bottom of which are collagen fibrils.[4] Biological[edit] In another context, structure can also observed in macromolecules, particularly proteins and nucleic acids.[6] The function of these molecules is determined by their shape as well as their composition, and their structure has multiple levels. Chemical[edit] Chemical structure refers to both molecular geometry and electronic structure. Mathematical[edit] Musical[edit] Social[edit] A social structure is a pattern of relationships. Data[edit] Software[edit] Logical[edit]
Universe (mathematics)
Collection that contains all the entities one wishes to consider in a given situation in mathematics The relationship between universe and complement In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In type theory, a universe is a type whose elements are types. Perhaps the simplest version is that any set can be a universe, so long as the object of study is confined to that particular set. Thus, even if the primary interest is X, the universe may need to be considerably larger than X. Then the superstructure over X, written SX, is the union of S0X, S1X, S2X, and so on; or So if the starting point is just X = {}, a great deal of the sets needed for mathematics appear as elements of the superstructure over {}. There is a slight shift in philosophy from the previous section, where the universe was any set U of interest.
Model theory
This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1] universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. and or
Union (set theory)
Set of elements in any of some sets Union of two sets: Union of three sets: The union of A, B, C, D, and E is everything except the white area. ) sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A = {x is an even integer larger than 1} B = {x is an odd integer larger than 1} As another example, the number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of even numbers {2, 4, 6, 8, 10, ...}, because 9 is neither prime nor even. Binary union is an associative operation; that is, for any sets Thus the parentheses may be omitted without ambiguity: either of the above can be written as for any set Also, the union operation is idempotent: All these properties follow from analogous facts about logical disjunction. Intersection distributes over union The power set of a set
Higher-order logic
Formal system of logic The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types (see Type theory). Quantification scope[edit] First-order logic quantifies only variables that range over individuals; second-order logic, in addition, also quantifies over sets; third-order logic also quantifies over sets of sets, and so on. Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply. Semantics[edit] There are two possible semantics for higher-order logic. In the standard or full semantics, quantifiers over higher-type objects range over all possible objects of that type. In Henkin semantics, a separate domain is included in each interpretation for each higher-order type. Properties[edit] See also[edit]
Uniformization (set theory)
From Wikipedia, the free encyclopedia In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if is a subset of , where and are Polish spaces, then there is a subset of that is a partial function from to such that exists) equals Such a function is called a uniformizing function for , or a uniformization of To see the relationship with the axiom of choice, observe that can be thought of as associating, to each element of , a subset of . then picks exactly one element from each such subset, whenever the subset is non-empty. A pointclass is said to have the uniformization property if every relation in can be uniformized by a partial function in . It follows from ZFC alone that have the uniformization property.
