New Foundations

In this section we usually consider NFU, which is known to be consistent, and discuss the effect of various "strong axioms of infinity". One can add such axioms as "there is an inaccessible cardinal" to NFU, but it is more natural to consider principles appropriate to this theory, usually assertions about cantorian and strongly cantorian sets: these are ways to strengthen the theory in its own terms, and they have the effect of causing large cardinals of the usual sorts to appear. NFU + Infinity + Choice, our usual base theory, has the same strength as TST + Infinity or Zermelo set t ...

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New Foundations, New Foundations - The type theory TST, New Foundations - Definition of New Foundations; stratification, New Foundations - Large sets appear, New Foundations - The consistency problem and related partial results, New Foundations - Resolving the paradoxes in NFU, New Foundations - Models of NFU, New Foundations - Self-sufficiency of mathematical foundations in NFU, New Foundations - Facts about the automorphism j, New Foundations - Strong Principles

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Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory. Some of the alternative set theories are: the theory of semisets; rough set theory; fuzzy set theory. the set theory New Foundations Positive set theory Specifically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his ...

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Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years after his first proof. His original argument did not mention decimal expansions, nor any other numeral system. Since this technique was first used, si ...

Including:

• Cantor's diagonal argument - Real numbers
• Cantor's diagonal argument - Why this does not work on integers
• Cantor's diagonal argument - General sets

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In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). See names of numbers in English. In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. While for finite sets the size is given by a natural number, the number of elements, cardinal numbers (cardinality ...

Including:

• Cardinal number - History
• Cardinal number - Motivation
• Cardinal number - Formal definition
• Cardinal number - Cardinal arithmetic
• Cardinal number - The continuum hypothesis

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In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. Universe mathematics - In a specific context. There are several precise versions of this general idea. Perhaps the simplest is that any set can be a universe, so long as you are studying that particular set. So if ...

Including:

• Universe mathematics - In a specific context
• Universe mathematics - In ordinary mathematics
• Universe mathematics - In set theory
• Universe mathematics - In category theory

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Willard Van Orman Quine (June 25, 1908 – December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. Willard Van Orman Quine - Overview. Quine falls squarely into the analytic philosophy tradition, though he is also the main proponent of the view that philosophy is not conceptual analysis. He served as the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 2000. His major writings include Two Dogmas of Empiricism, which influentially ...

Including:

• Willard Van Orman Quine - Overview
• Willard Van Orman Quine - Life
• Willard Van Orman Quine - Work
• Willard Van Orman Quine - Rejection of the analytic-synthetic distinction
• Willard Van Orman Quine - The indeterminacy of translation
• Willard Van Orman Quine - Confirmation holism and ontological relativity
• Willard Van Orman Quine - Set Theory Mathematical Logic
• Willard Van Orman Quine - The teaching of formal logic
• Willard Van Orman Quine - Quotations
• Willard Van Orman Quine - Notable books by Quine
• Willard Van Orman Quine - Literature about Quine
• Willard Van Orman Quine - Quine in Popular Media

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Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

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A formula φ in the language of first-order logic with equality and membership is said to be stratified for the purposes of New Foundations and related set theories iff there is a function σ which sends each variable appearing in φ (considered as an item of syntax) to a natural number (this works equally well if all integers are used) in such a way that any atomic formula appearing in φ satisfies See also:

Stratification mathematics, Stratification mathematics - Stratification in mathematical logic, Stratification mathematics - Stratification in New Foundations, Stratification mathematics - Stratification in singularity theory

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The Barber paradox, in addition to leading to a tidier set theory, has been used twice more with great success: Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of the Halting problem (and with that the Entscheidungsproblem) by using the same trick. Russell's paradox - Russell-like paradoxes. As illustrated above for Barbers and Lists, the Russell paradox is not hard to extend. Needed is A transitive verb < ...

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Russell's paradox, Russell's paradox - History, Russell's paradox - Applied versions, Russell's paradox - Set-theoretic responses, Russell's paradox - Responses illustrated, Russell's paradox - Applications and related topics, Russell's paradox - Russell-like paradoxes, Russell's paradox - Reciprocation, Russell's paradox - Independence from excluded middle, Russell's paradox - Other related paradoxes

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However, once you consider subsets of a given set X (in Cantor's case, X = R), you may become interested in sets of subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will not themselves be subsets of X but will instead be subsets of PX, the power set of X. Of course, it doesn't stop there; you might next be interested in sets of sets of subsets of X, and so on. In another direction, you may become interest ...

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Universe mathematics, Universe mathematics - In a specific context, Universe mathematics - In ordinary mathematics, Universe mathematics - In set theory, Universe mathematics - In category theory

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Cantor's original proof shows that the interval [0,1] is not countably infinite. The proof by contradiction proceeds as follows: Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in orde ...

See also:

Cantor's diagonal argument, Cantor's diagonal argument - Real numbers, Cantor's diagonal argument - Why this does not work on integers, Cantor's diagonal argument - General sets

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In informal use, a cardinal number is what is normally referred to as a counting number. They may be identified with the natural numbers beginning with 0 (i.e. 0, 1, 2, ...). The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic. More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is e ...

See also:

Cardinal number, Cardinal number - History, Cardinal number - Motivation, Cardinal number - Formal definition, Cardinal number - Cardinal arithmetic, Cardinal number - The continuum hypothesis

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The axiom schema of separation can almost be derived from the axiom schema of replacement. First, recall this axiom schema: for any functional predicate F in one variable that doesn't use the symbols A, B, C or D. Given a suitable predicate P for the axiom of specification, define the mapping F by F(D) = D if P(D) is true and F(D) = E if P(D) is false, where E is any member of ASee also:

Axiom schema of specification, Axiom schema of specification - Relation to the axiom schema of replacement, Axiom schema of specification - Unrestricted comprehension, Axiom schema of specification - In NBG class theory, Axiom schema of specification - In second order logic, Axiom schema of specification - In Quine's New Foundations

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The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to von Neumann under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti. Here is an account with fewer presuppositions: suppose that we associate with each well-ordering an object called its "order type" in an unspecified way (the order types are the ordinal numbers). The "order types" (ordinal numbers) themselves are well-ordered in a natural way, and this w ...

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Burali-Forti paradox, Burali-Forti paradox - Stated in terms of von Neumann ordinals, Burali-Forti paradox - Stated more generally, Burali-Forti paradox - Resolution of the paradox in ZFC, Burali-Forti paradox - History

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Quine's Ph.D. thesis and early publications were on formal logic and set theory. Only after WWII did he emerge as a major philosopher, by virtue of seminar papers on ontology, epistemology, and language, By the 1960s he had worked out his "naturalized epistemology," whose aim was to answer all substantive questions of knowledge and meaning using the methods and tools of the natural sciences. Quine roundly rejected the notion that there should be a "first philosophy," a theoretical standpoint somehow prior to natural science, and capable of justifying ...

See also:

Willard Van Orman Quine, Willard Van Orman Quine - Overview, Willard Van Orman Quine - Life, Willard Van Orman Quine - Work, Willard Van Orman Quine - Rejection of the analytic-synthetic distinction, Willard Van Orman Quine - Confirmation holism and ontological relativity, Willard Van Orman Quine - Set Theory, Willard Van Orman Quine - The logic and mathematics teacher, Willard Van Orman Quine - Quotations, Willard Van Orman Quine - Notable books by Quine, Willard Van Orman Quine - Literature about Quine, Willard Van Orman Quine - Quine in Popular Culture

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The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...

See also:

Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

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The original set theoretical definition of the natural numbers is generally ascribed to Frege and Russell. An informal way to put this definition is that each concrete natural number n is defined as the set of all sets with n elements. This appears circular but is not; it is necessary to be more explicit to see this. Define 0 as {{}} (obviously the set of all sets with 0 elements). For any set A, define σ(A) as , the set of all sets obtainable by adding one new element ...

See also:

Set-theoretic definition of natural numbers, Set-theoretic definition of natural numbers - The oldest definition, Set-theoretic definition of natural numbers - The usual definition

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topology on the classes. The theory can interpret ZFC (by restricting oneself to the class of well-founded sets, which is not itself a set). It in fact interprets a stronger theory (Morse-Kelley set theory with the proper class ordinal a weakly compact cardinal). ...

See also:

Positive set theory, Positive set theory - Interesting properties, Positive set theory - Researchers

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In Logic (1953), Rosser uses a different definition (ultimately due to Willard van Orman Quine), one which requires that a definition of the natural numbers be in place. Let Nn be the set of natural numbers, and define That is, φ(x) contains the successor of every natural number in x, together with all the non-numbers from x. In particular, it does not contain the number 0, so that for any sets A and B, . Then the ordered pair (A,B) may be defined by adjoining 0 to each element of φ(B), and uniting ...

See also:

Ordered pair, Ordered pair - Proofs of the characteristic property of ordered pairs, Ordered pair - Rosser definition, Ordered pair - Morse definition

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To define the sum S + T of two ordinal numbers S and T, one proceeds as follows: first the elements of T are relabeled so that S and T become disjoint, then the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S∪T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This way, a new well-ordered set is formed, ...

See also:

Ordinal number, Ordinal number - Introduction, Ordinal number - The oldest definition, Ordinal number - Modern definition and first properties, Ordinal number - Other definitions, Ordinal number - Arithmetic of ordinals, Ordinal number - Cantor normal form, Ordinal number - Topology and limit ordinals

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