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Naive Set Theory (book)
:''See also Naive set theory for the mathematical topic.'' ''Naive Set Theory'' is a mathematics textbook by Paul Halmos providing an undergraduate introduction to set theory. Originally published by ''Van Nostrand'' in 1960, it was reprinted in the Springer-Verlag Undergraduate Texts in Mathematics series in 1974. It is on the list of 173 books essential for undergraduate math libraries. While the title states that the set theory presented is 'naive', which is usually taken to mean without formal axioms, the book does introduce a system of axioms equivalent to that of ZFC set theory except the Axiom of foundation. It also gives correct and rigorous definitions for many basic concepts.Review of ''Naive Set Theory'', L. Rieger, . Where it differs from a "true" axiomatic set theory book is its character: there are no discussions of axiomatic minutiae, and there is next to nothing about advanced topics such as large cardinals or forcing. Instead, it tries to be intelligible t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Halmos
Paul Richard Halmos (; 3 March 1916 – 2 October 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and probabilist who made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians (scientists), The Martians. Early life and education Born in the Kingdom of Hungary into a History of the Jews in Hungary, Jewish family, Halmos immigrated to the United States at age 13. He obtained his B.A. from the University of Illinois at Urbana-Champaign, University of Illinois, majoring in mathematics while also fulfilling the requirements for a degree in philosophy. He obtained the degree after only three years, and was 19 years old when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus. However, after failin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Schema Of Specification
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Union
An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so Self-evidence, evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a "#Logical axioms, logical axiom" or a "#Non-logical axioms, non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton (mathematics)
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and \ are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as \ is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton \. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of \, which is the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Pairing
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary sets. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\forall A \, \forall B \, \exists C \, \forall D \, \in C \iff (D = A \lor D = B)/math> In words: :Given any object ''A'' and any object ''B'', there is a set ''C'' such that, given any object ''D'', ''D'' is a member of ''C'' if and only if ''D'' is equal to ''A'' or ''D'' is equal to ''B''. Consequences As noted, what the axiom is saying is that, given two objects ''A'' and ''B'', we can find a set ''C'' whose members are exactly ''A'' and ''B''. We can use the axiom of extensionality to show that this set ''C'' is unique. We call the set ''C'' the ''pair'' of ''A'' and ''B'', and denote it . Thus the essence of the axiom is: :Any tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called ''non-empty''. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø () in the Danish orthography, Danish and Norwegian orthography, Norwegian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Formula
In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives. The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a first-order logic#Formation rules, term. In model theory, atomic formulas are merely string (computer science), strings of symbols with a given signature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Quantification
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("", "", or sometimes by "" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). The universal quantifier is en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Quantification
Existentialism is a family of philosophy, philosophical views and inquiry that explore the human individual's struggle to lead an Authenticity (philosophy), authentic life despite the apparent Absurdity#The Absurd, absurdity or incomprehensibility of existence. In examining meaning of life, meaning, purpose, and value (ethics), value, existentialist thought often includes concepts such as existential crisis, existential crises, Angst#Existentialist angst, angst, courage, and freedom. Existentialism is associated with several 19th- and 20th-century European philosophers who shared an emphasis on the human subject, despite often profound differences in thought. Among the 19th-century figures now associated with existentialism are philosophers Søren Kierkegaard and Friedrich Nietzsche, as well as novelist Fyodor Dostoevsky, all of whom critiqued rationalism and concerned themselves with the problem of meaning (philosophy), meaning. The word ''existentialism'', however, was not coin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first-order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The most commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negation
In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. For example, if P is "Spot runs", then "not P" is "Spot does not run". An operand of a negation is called a ''negand'' or ''negatum''. Negation is a unary operation, unary logical connective. It may furthermore be applied not only to propositions, but also to notion (philosophy), notions, truth values, or interpretation (logic), semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Disjunction
In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula S \lor W , assuming that S abbreviates "it is sunny" and W abbreviates "it is warm". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
