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Scott–Potter Set Theory
An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott and the philosopher George Boolos. Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmetic and the other usual number systems, and the theory of relations. ZU etc. Preliminaries This section and the next follow Part I of Potter (2004) closely. The background logic is first-order logic with identity. The ontology includes urelements as well as sets, which makes it clear that there can be sets of entities defined by first-order theories not based on sets. The urelements are not essential in that other mathematical structures can be defined as sets, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theorems, proof (mathematics), proofs, algorithms, etc. in particular. This may also include the philosophy of mathematics, philosophical study of the relation of this framework with reality. The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's Elements, Euclid's ''Elements''. A mathematical assertion is considered as truth (mathematics), truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms (inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Quantification
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols " ∃!" or "∃=1". It is defined to mean there exists an object with the given property, and all objects with this property are equal. For example, the formal statement : \exists! n \in \mathbb\,(n - 2 = 4) may be read as "there is exactly one natural number n such that n - 2 =4". Proving uniqueness The most common technique to prove the unique existence of an object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, ''a'' and ''b'') must be equal to each other (i.e. a = b). For example, to show that the equation x + 2 = 5 has exactly one solution, one would first start by establishing that at least one solution exists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of A Function
In mathematics, the domain of a function is the Set (mathematics), set of inputs accepted by the Function (mathematics), function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be". More precisely, given a function f\colon X\to Y, the domain of is . In modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both sets of real numbers, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the ''codomain'': the set to which all outputs must belong. The set of specific outputs the function assigns to elements of is called its ''Range of a function, range'' or ''Image (mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...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] |
Inductive Set (axiom Of Infinity)
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908. Formal statement Using first-order logic primitive symbols, the axiom can be expressed as follows: \exist \mathrm \ (\exist o \ (o \in \mathrm \ \land \lnot \exist n \ (n \in o)) \ \land \ \forall x \ (x \in \mathrm \Rightarrow \exist y \ (y \in \mathrm \ \land \ \forall a \ (a \in y \Leftrightarrow (a \in x \ \lor \ a = x))))). If the notations of both set-builder and empty set are allowed: \exists \mathrm \, ( \varnothing \in \mathrm \, \land \, \forall x \, (x \in \mathrm \Rightarrow \, ( x \cup \ ) \in \mathrm ) ). Some mathematicians may call a set built this way an inductive set. Hint: In English, it reads: "There exists a set 𝐈 suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite
Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * Finite number (other) * Finite part (other) * Finite map (other) * Finite presentation (other) * Finite type (other) Finite type refers to several related concepts in mathematics: * Algebra of finite type, an associative algebra with finitely many generators **Morphism of finite type, a morphism of schemes with underlying morphisms on affine opens given by algebr ... * * Nonfinite (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Actual Infinite
In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves infinite entities as given, actual and completed objects. The concept of actual infinity was introduced into mathematics near the end of the 19th century by Georg Cantor with his theory of infinite sets, and was later formalized into Zermelo–Fraenkel set theory. This theory, which is presently commonly accepted as a foundation of mathematics, contains the axiom of infinity, which means that the natural numbers form a set (necessarily infinite). A great discovery of Cantor is that, if one accepts infinite sets, then there are different sizes ( cardinalities) of infinite sets, and, in particular, the cardinal of the continuum of the real numbers is strictly larger than the cardinal of the natural numbers. Actual infinity is to be contrasted with potential infinity, in which an endless process (such as "add 1 to the previous number") produces a sequence with no last el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908. Formal statement Using first-order logic primitive symbols, the axiom can be expressed as follows: \exist \mathrm \ (\exist o \ (o \in \mathrm \ \land \lnot \exist n \ (n \in o)) \ \land \ \forall x \ (x \in \mathrm \Rightarrow \exist y \ (y \in \mathrm \ \land \ \forall a \ (a \in y \Leftrightarrow (a \in x \ \lor \ a = x))))). If the notations of both set-builder and empty set are allowed: \exists \mathrm \, ( \varnothing \in \mathrm \, \land \, \forall x \, (x \in \mathrm \Rightarrow \, ( x \cup \ ) \in \mathrm ) ). Some mathematicians may call a set built this way an inductive set. Hint: In English, it reads: " There exists a set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Schema Of Separation
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (''Aussonderungsaxiom''), subset axiom, axiom of class construction, 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 \varphi in the language of set theory with free variables among ''x'', ''w''1, ..., ''w''''n'', ''A''. So ''B'' does not occur free in \varphi. In the formal language of set theory, the axiom schema is: :\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
