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Axiom Of Adjunction
In mathematical set theory, the axiom of adjunction states that for any two sets ''x'', ''y'' there is a set ''w'' = ''x'' ∪ given by "adjoining" the set ''y'' to the set ''x''. : \forall x \,\forall y \,\exists w \,\forall z\, z \in w \leftrightarrow (z \in x \lor z=y) introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as general set theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursive set functions. Tarski and Smielew showed that Robinson arithmetic (Q) can be interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction . In fact, empty set and adjunction alone (without extensionality) suffice to interpret Q. (They are mutually interpretable.) Adding epsilon-induction to empty set and adjunction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Set Theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. Ontology The ontology of GST is identical to that of ZFC, and hence is thoroughly canonical. GST features a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (hence all mathematical objects) are sets. There is a single primitive binary relation, set membership; that set ''a'' is a member of set ''b'' is written ''a ∈ b'' (usually read "''a'' is an element of ''b''"). Axioms The symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. As with Z, the background logic for GST is first order logic with identity. Indeed, GST is the fragment of Z obtained by omitting the axioms Union, Power Set, Elementary Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitary Set Theory
In mathematics and logic, an Operation (mathematics), operation is finitary if it has Finite cardinality, finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an Infinite set, infinite number of input values. In standard mathematics, an operation is finitary by definition. Therefore these terms are usually only used in the context of infinitary logic. Finitary argument A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finiteThe number of axioms ''referenced'' in the argument will necessarily be finite since the proof is finite, but the number of axioms from which these are ''chosen'' is infinite when the system has axiom schemes, e.g. the axiom schemes of propositional calculus. set of axioms. In other words, it is a Mathematical proof, proof (including all assumptions) that can be written on a large enough sheet of paper. By contrast, infinitary logic studies lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Recursive Set Function
In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by . Definition A primitive recursive set function is a function from sets to sets that can be obtained from the following basic functions by repeatedly applying the following rules of substitution and recursion: The basic functions are: *Projection: ''P''''n'',''m''(''x''1, ..., ''x''''n'') = ''x''''m'' for 0 ≤ ''m'' ≤ ''n'' *Zero: ''F''(''x'') = 0 * Adjoining an element to a set: ''F''(''x'', ''y'') = ''x'' ∪ *Testing membership: ''C''(''x'', ''y'', ''u'', ''v'') = ''x'' if ''u'' ∈ ''v'', and ''C''(''x'', ''y'', ''u'', ''v'') = ''y'' otherwise. The rules for generating new functions by substitution are *''F''(x, y) = ''G''(x, ''H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy. Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983. Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robinson Arithmetic
In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable. Axioms The background logic of Q is first-order logic with identity, denoted by infix '='. The individuals, called natural numbers, are members of a set called N with a distinguished member 0, called zero. There are three operations over N: *A unary operation called successor and denoted by prefix ''S''; *Two binary operations, addition and multiplication, denoted by infix + and ·, respectively. The following axioms for Q are Q1–Q7 in (cf. also the axioms of first-order arithmetic). Variables not bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epsilon-induction
In set theory, \in-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion. It may also be studied in a general context of induction on well-founded relations. Statement The schema is for any given property \psi of sets and states that, if for every set x, the truth of \psi(x) follows from the truth of \psi for all elements of x, then this property \psi holds for all sets. In symbols: :\forall x. \Big(\big(\forall (y \in x). \psi(y)\big)\,\to\,\psi(x)\Big)\,\to\,\forall z. \psi(z) Note that for the "bottom case" where x denotes the empty set \, the subexpression \forall(y\in x).\psi(y) is vacuously true for all propositions and so that implication is proven by just proving \psi(\). In words, if a property is persistent when collecting any sets with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithmetice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Separation
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set. Some mathematicians call it the axiom schema of comprehension, although others use that term for ''unrestricted'' comprehension, discussed below. Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory. Statement One instance of the schema is included for each formula φ in the language of set theory with free variables among ''x'', ''w''1, ..., ''w''''n'', ''A''. So ''B'' does not occur free in φ. In the formal language of set theory, the axiom schema is: :\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow x \in A \land \varphi(x, w_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Schema
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term. Finite axiomatization Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. A theory that can be axiomatized without schemata is said to be finitely axiomatized. Theories that can be finitely axiomatized are seen as a bit more metamathematically elegant, even if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
