Internal set theory

There are three axioms of IST to add to the established ZFC set theoretic axioms (note that use of the ZFC axiom schemas is restricted: the axiom schemas of separation and replacement can only be used with classical formulas, just as in ZFC proper) - conveniently one for each letter in the name: Idealisation, Standardisation, and Transfer. All the principles described above can b ...

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Internal set theory, Internal set theory - Intuitive justification, Internal set theory - Principles of the standard predicate, Internal set theory - Formal axioms for IST, Internal set theory - I : Idealisation, Internal set theory - S : Standardisation, Internal set theory - T : Transfer, Internal set theory - Formal justification for the axioms

Read more here: » Internal set theory: Encyclopedia II - Internal set theory - Formal axioms for IST

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

Read more here: » Axiomatic set theory: Encyclopedia - Axiomatic set theory

There are at least three reasons to consider non-standard analysis: Non-standard analysis - Historical. Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity. As noted in the article on hyperreal numbers, these formulations were widely criticized by Bishop Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and it is arguable that the first person to solve this in a satisfa ...

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Non-standard analysis, Non-standard analysis - Motivation, Non-standard analysis - Historical, Non-standard analysis - Pedagogical, Non-standard analysis - Technical, Non-standard analysis - Approaches to non-standard analysis, Non-standard analysis - Applications, Non-standard analysis - Applications to calculus, Non-standard analysis - Criticisms, Non-standard analysis - Logical framework, Non-standard analysis - First consequences

Read more here: » Non-standard analysis: Encyclopedia II - Non-standard analysis - Motivation

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 ...

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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

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory

Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B. The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B. Finally, the relative complement of B relative to A, also known as the set theoretic differenc ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Unions intersections and relative complements

Despite the elegance and appeal of some aspects of non-standard analysis, there is a great deal of skepticism in the mathematical community about whether this machinery really adds anything that cannot just as easily be achieved by standard methods. One noted critic of non-standard analysis is the Fields Medalist Alain Connes, as evinced by the following quote The answer given by nonstandard analysis, a so-called nonstandard real, is equally deceiving. From every nonstandard real number one can construct canonically a sub ...

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Non-standard analysis, Non-standard analysis - Motivation, Non-standard analysis - Historical, Non-standard analysis - Pedagogical, Non-standard analysis - Technical, Non-standard analysis - Approaches to non-standard analysis, Non-standard analysis - Applications, Non-standard analysis - Applications to calculus, Non-standard analysis - Criticisms, Non-standard analysis - Logical framework, Non-standard analysis - First consequences

Read more here: » Non-standard analysis: Encyclopedia II - Non-standard analysis - Criticisms

Despite some initial hope in the mathematical community that non-standard analysis would alter the way mathematicians thought about and reasoned with real numbers, this expectation never materialized. Moreover the list of new applications in mathematics is still very small. One of these results is the theorem proven by Abraham Robinson and Allen Bernstein that every polynomially compact linear operator on a Hilbert space has an invariant subspace. Upon reading a preprint of the Bernstein-Robinson paper, Paul Halmos reinterpreted their proof ...

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Non-standard analysis, Non-standard analysis - Motivation, Non-standard analysis - Historical, Non-standard analysis - Pedagogical, Non-standard analysis - Technical, Non-standard analysis - Approaches to non-standard analysis, Non-standard analysis - Applications, Non-standard analysis - Applications to calculus, Non-standard analysis - Criticisms, Non-standard analysis - Logical framework, Non-standard analysis - First consequences

Read more here: » Non-standard analysis: Encyclopedia II - Non-standard analysis - Applications

There are two very different approaches to non-standard analysis: the semantic or model-theoretic approach and the syntactic approach. Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology. The semantic approach is by far the most popular approach to non-standard analysis. Robinson's original formulation of non-standard analysis falls into this category. As developed by him in his papers, it is based on studying models (in particular saturated models) of a theor ...

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Non-standard analysis, Non-standard analysis - Motivation, Non-standard analysis - Historical, Non-standard analysis - Pedagogical, Non-standard analysis - Technical, Non-standard analysis - Approaches to non-standard analysis, Non-standard analysis - Applications, Non-standard analysis - Applications to calculus, Non-standard analysis - Criticisms, Non-standard analysis - Logical framework, Non-standard analysis - First consequences

Read more here: » Non-standard analysis: Encyclopedia II - Non-standard analysis - Approaches to non-standard analysis

Given any set S, the superstructure over a set S is the set V(S) defined by the conditions Thus the superstructure over S is obtained by starting from S and iterating the operation of adjoining the power set of S and taking the union of the resulting sequence. The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it contains isomorphic copies of all separable metric spaces and metrizable topological vector spaces. Virtually all of mathematics t ...

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Non-standard analysis, Non-standard analysis - Motivation, Non-standard analysis - Historical, Non-standard analysis - Pedagogical, Non-standard analysis - Technical, Non-standard analysis - Approaches to non-standard analysis, Non-standard analysis - Applications, Non-standard analysis - Applications to calculus, Non-standard analysis - Criticisms, Non-standard analysis - Logical framework, Non-standard analysis - First consequences

Read more here: » Non-standard analysis: Encyclopedia II - Non-standard analysis - Logical framework

The symbol *N denotes the nonstandard natural numbers. By the extension principle, this is a superset of N. The set *N − N is nonempty. To see this, apply countable saturation to the sequence of internal sets The sequence {Ak}k ∈ N has a nonempty intersection, proving the result. We begin with some definitions: Hyperreals r ...

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Non-standard analysis, Non-standard analysis - Motivation, Non-standard analysis - Historical, Non-standard analysis - Pedagogical, Non-standard analysis - Technical, Non-standard analysis - Approaches to non-standard analysis, Non-standard analysis - Applications, Non-standard analysis - Applications to calculus, Non-standard analysis - Criticisms, Non-standard analysis - Logical framework, Non-standard analysis - First consequences

Read more here: » Non-standard analysis: Encyclopedia II - Non-standard analysis - First consequences

The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC. An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, s ...

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

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Axioms for set theory

Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can ...

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

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Independence in ZFC

From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function , etc. For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair ( a, b ) which represents the pairing of two objects in this order. The defining property of an ordered pair is that ( a, b ) = ( c, d ) if and only if a = c and b = d. The approach is basically to specify th ...

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

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics

We referred earlier to the need for a formal, axiomatic approach. What problems arise in the treatment we have given? The problems relate to the formation of sets. One's first intuition might be that we can form any sets we want, but this view leads to inconsistencies. For any set x we can ask whether x is a member of itself. Define Z = {x : x is not a member of x}. Now for the problem: is Z a member of Z? If yes, then by the defining quality of Z ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Paradoxes

The simplest way to describe a set is to list its elements between curly braces. Thus {1,2} denotes the set whose only elements are 1 and 2. (See axiom of pairing.) Note the following points: Order of elements is immaterial; for example, {1,2} = {2,1}. Repetition (multiplicity) of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}. (These are consequences of the definition of equality in the previous section.) This notation can be informally abused by saying something like {dogs} to indicate the set of all dogs, but this example would usually be read by mathematicians as ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Specifying sets

In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. As can be seen from this example, sets are allowed to have an infinite number of elements. If x is a member of A, then it is also said that x belongs to A, or that x is in A. In this case, we write x&# ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Sets membership and equality

Naive set theory was created at the end of the 19th century by Georg Cantor in order to allow mathematicians to work with infinite sets consistently. As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox. In response, axiomatic set theory was developed to determine precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually mean axiomatic set theory, but when they talk about set theory as a mere tool to be applied to other mathema ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Introduction

Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set th ...

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

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Objections to set theory

Given two sets A and B we say that A is a subset of B if every element of A is also an element of B. Notice that in particular, B is a subset of itself; a subset of B that isn't equal to B is called a proper subset. If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Subsets

Intuitively, an ordered pair is simply a collection of two objects such that one can be distinguished as the first element and the other as the second element, and having the fundamental property that, two ordered pairs are equal if and only if their first elements are equal and their second elements are equal. Formally, an ordered pair with first coordinate a, and second coordinate b, usually denoted by (a, ...

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Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote

Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Ordered pairs and Cartesian products