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Subcountable
In constructive mathematics, a collection X is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as \exists (I\subseteq).\, \exists f.\, (f\colon I\twoheadrightarrow X), where f\colon I\twoheadrightarrow X denotes that f is a surjective function from I onto X. The surjection is a member of \rightharpoonup X and here the subclass I of is required to be a set. In other words, all elements of a subcountable collection X are functionally in the image of an indexing set of counting numbers I\subseteq and thus the set X can be understood as being dominated by the countable set . Discussion Nomenclature Note that nomenclature of countability and finiteness properties vary substantially - in part because many of them coincide when assuming excluded middle. To reiterate, the discussion here concerns the property defined in terms of surjections onto the set X being characterized. The language here is common in constructive set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor's Diagonal Argument
Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbersinformally, that there are sets which in some sense contain more elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began. Georg Cantor published this proof in 1891, English translation: but it was not his first proof of the uncountability of the real numbers, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the ''Entscheidungsproblem''. Diagonalization arguments ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Luitzen Egbertus Jan Brouwer, Brouwer, the finitism of David Hilbert, Hilbert and Paul Bernays, Bernays, the constructive recursive mathematics of Nikolai Aleksandrovich Shanin, Shanin and Andrey Markov (Soviet mathematician), Markov, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Logic
Constructive logic is a family of logics where proofs must be constructive (i.e., proving something means one must build or exhibit it, not just argue it “must exist” abstractly). No “non-constructive” proofs are allowed (like the classic proof by contradiction without a witness). The main constructive logics are the following: 1. Intuitionistic logic Founder: L. E. J. Brouwer (1908, philosophy) formalized by A. Heyting (1930) and A. N. Kolmogorov (1932) Key Idea: Truth = having a proof. One cannot assert “P or not P” unless one can prove P or prove \neg \neg P. Features: * No law of excluded middle (P \lor \neg P is not generally valid). * No double negation elimination (\neg \neg\ P \to P is not valid generally). * Implication is constructive: a proof of P \to Q is a method turning any proof of P into a proof of Q. Used in: type theory, constructive mathematics. 2. Modal logics for constructive reasoning Founder(s): * K F. Gödel (1933) showed that intui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Schema Of Replacement
In set theory, the axiom schema of replacement is a Axiom schema, schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image (mathematics), image of any Set (mathematics), set under any definable functional predicate, mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a class (set theory), class is a set depends only on the cardinality of the class, not on the rank (set theory), rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining Well-formed formula, formulas. Statement Suppose P is a definable binary relation (mathematics), relation (which may be a proper class) such that f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain of . If equals , that is, if is defined on every element in , then is said to be a total function. In other words, a partial function is a binary relation over two sets that associates to every element of the first set ''at most'' one element of the second set; it is thus a univalent relation. This generalizes the concept of a (total) function by not requiring ''every'' element of the first set to be associated to an element of the second set. A partial function is often used when its exact domain of definition is not known, or is difficult to specify. However, even when the exact domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in calculus, where, for example, the quotien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computably Enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an enumeration algorithm, algorithm that enumerates the members of ''S''. That means that its output is a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever, but each element of S will be returned after a finite amount of time. Note that these elements do not have to be listed in a particular way, say from smallest to largest. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm can run forever, and no inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Of A Function
In mathematics, the range of a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called ''surjective'' or ''onto''. For any non-surjective function f: X \to Y, the codomain Y and the image \tilde Y are different; however, a new function can be defined with the original function's image as its codomain, \tilde: X \to \tilde where \tilde(x) = f(x). This new function is surjective. Definitions Given two sets and , a binary relation between and is a function (from to ) if for every element in there is exactly one in such that relates to . The sets and are called the '' domain'' and ''codomain'' of , respectively. The ''image'' of the function is the subset of consisting of only those elements of such that there is at least one in with . Usage As the term "range" can have different meanings, it is considered a good practice to define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Creative And Productive Sets
In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic. They are a standard topic in mathematical logic textbooks such as and . Definition and example For the remainder of this article, assume that \varphi_i is an admissible numbering of the computable functions and ''W''''i'' the corresponding numbering of the recursively enumerable sets. A set ''A'' of natural numbers is called productive if there exists a total recursive (computable) function f so that for all i \in \mathbb, if W_i \subseteq A then f(i) \in A \setminus W_i. The function f is called the productive function for A. A set ''A'' of natural numbers is called creative if ''A'' is recursively enumerable and its complement \mathbb\setminus A is productive. Not every productive set has a recursively enumerable complement, however, as illustrated below. The archetypal creative set is K = \, the set representing the halting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church's Thesis (constructive Mathematics)
In constructive mathematics, Church's thesis is the principle stating that all total functions are computable functions. The similarly named Church–Turing thesis states that every '' effectively calculable function'' is a ''computable function'', thus collapsing the former notion into the latter. is stronger in the sense that with it every function is computable. The constructivist principle is however also given, in different theories and incarnations, as a fully formal axiom. The formalizations depends on the definition of "function" and "computable" of the theory at hand. A common context is recursion theory as established since the 1930's. Adopting as a principle, then for a predicate of the form of a family of existence claims (e.g. \exists! y. \varphi(x,y) below) that is proven not to be validated for all x in a computable manner, the contrapositive of the axiom implies that this is then not validated by ''any'' total function (i.e. no mapping corresponding to x\mapsto y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countably Infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Excluded Middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
