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Axiomatic set theory | A Wisdom Archive on Axiomatic set theory | | Axiomatic set theory A selection of articles related to Axiomatic set theory | |
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Axiomatic set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Objections to set theory, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Well-foundedness and hypersets, Alternative set theory, List of set theory topics, Zermelo-Fraenkel set theory, Simple theorems in the algebra of sets, Naive set theory, Cantor–Bernstein–Schroeder theorem, Zorn's lemma, Cantor's theorem, Cantor's diagonal argument, Model theory, Internal set theory, Kripke-Platek set theory with urelements
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| ARTICLES RELATED TO Axiomatic set theory | | | |  Axiomatic set theory: Encyclopedia II - Naive set theory - Ordered pairs and Cartesian productsIntuitively, 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, ...
See also: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 |
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| | | | | Axiomatic set theory: Encyclopedia II - Naive set theory - Paradoxes 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 ...
See also: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 |
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| | | | | Axiomatic set theory: Encyclopedia II - Theory - Science In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it often does in other contexts. Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, or conjectures. A theory is in this context a set of hypotheses that are logically bound together (See also hypothetico-deductive method).
Theories are typically ways of explaining why things happen, often, but not always after their occurrence is no longer in scientific di ...
See also:Theory, Theory - Etymology, Theory - Science, Theory - Models, Theory - Types of theories, Theory - Further explanation of a scientific theory, Theory - Characteristics, Theory - Mathematics, Theory - Other fields, Theory - List of famous theories, Theory - Reference Read more here: » Theory: Encyclopedia II - Theory - Science |
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| | | | | | Axiomatic set theory: Encyclopedia II - Mathematical logic - Technical reference
Mathematical logic - First-order languages and structures.
Definition. A first-order language is a collection of distinct typographical symbols classified as follows:
The equality symbol ; the connectives , ; the universal quantifier and the parentheses , .
A countable set of variable symbols .
A set of constant symbols .
A set of function symbol ...
See also:Mathematical logic, Mathematical logic - History, Mathematical logic - Topics in mathematical logic, Mathematical logic - Some fundamental results, Mathematical logic - Technical reference, Mathematical logic - First-order languages and structures, Mathematical logic - Terms formulas and sentences, Mathematical logic - Assignment functions, Mathematical logic - Logical satisfaction, Mathematical logic - Logical implication and truth, Mathematical logic - Variable substitution, Mathematical logic - Substitutability Read more here: » Mathematical logic: Encyclopedia II - Mathematical logic - Technical reference |
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| | | | | Axiomatic set theory: Encyclopedia II - Theory - Science In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it does in other contexts. Neither is a scientific theory a fact. Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, theories, or laws.
Theories are typically ways of explaining why things happen, often, but not always after their occurrence is no longer in scientific dispute. In referring to the "theory of global warming" for example, the worldwide ...
See also:Theory, Theory - Etymology, Theory - Science, Theory - Models, Theory - Types of theories, Theory - Further explanation of a scientific theory, Theory - Characteristics, Theory - Mathematics, Theory - Other fields, Theory - List of famous theories, Theory - Reference Read more here: » Theory: Encyclopedia II - Theory - Science |
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| | | | | Axiomatic set theory: Encyclopedia II - Mathematical logic - Topics in mathematical logic The main areas of mathematical logic include model theory, proof theory and recursion theory (often now referred to as computability theory). Axiomatic set theory is sometimes considered too. There are many overlaps with computer science, since many early pioneers in computer science, such as Alan Turing, were mathematicians and logicians.
The study of programming language semantics derives from model ...
See also:Mathematical logic, Mathematical logic - History, Mathematical logic - Topics in mathematical logic, Mathematical logic - Some fundamental results, Mathematical logic - Technical reference, Mathematical logic - First-order languages and structures, Mathematical logic - Terms formulas and sentences, Mathematical logic - Assignment functions, Mathematical logic - Logical satisfaction, Mathematical logic - Logical implication and truth, Mathematical logic - Variable substitution, Mathematical logic - Substitutability Read more here: » Mathematical logic: Encyclopedia II - Mathematical logic - Topics in mathematical logic |
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| | | | | Axiomatic set theory: Encyclopedia II - Theory - Mathematics In mathematics, the word theory is used informally to refer to certain distinct bodies of knowledge about mathematics. This knowledge consists of axioms, definitions, theorems and computational techniques, all related in some way by tradition or practice. Examples include group theory, set theory, Lebesgue integration theory and field theory.
The term "theory" also has a formal usage in mathematics, particularly in mathematical logic and model theory. A theory in this sense is a set of statements closed under certain rul ...
See also:Theory, Theory - Etymology, Theory - Science, Theory - Models, Theory - Types of theories, Theory - Further explanation of a scientific theory, Theory - Characteristics, Theory - Mathematics, Theory - Other fields, Theory - List of famous theories, Theory - Reference Read more here: » Theory: Encyclopedia II - Theory - Mathematics |
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| | | | | Axiomatic set theory: Encyclopedia II - Mathematical logic - History Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra.
Attempts to treat the operations of formal logic in a symbolic or algebraic way were made by some of the more philosophical mathematicians, such as Leibniz and Lambert; but their labors remained little known and isolated. It was George Boole and then Augustus De Morgan, in the middle of the ninetee ...
See also:Mathematical logic, Mathematical logic - History, Mathematical logic - Topics in mathematical logic, Mathematical logic - Some fundamental results, Mathematical logic - Technical reference, Mathematical logic - First-order languages and structures, Mathematical logic - Terms formulas and sentences, Mathematical logic - Assignment functions, Mathematical logic - Logical satisfaction, Mathematical logic - Logical implication and truth, Mathematical logic - Variable substitution, Mathematical logic - Substitutability Read more here: » Mathematical logic: Encyclopedia II - Mathematical logic - History |
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| | | | | Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Objections to set theory 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 |
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| | | | | Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness:
a set, x0, is well founded iff it has no infinite descending membership sequence:
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In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC wit ...
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 - Well-foundedness and hypersets |
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| | | | | Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Axioms for set theory 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 |
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| | | | | Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Independence in ZFC 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 |
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| | | | | Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics 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 |
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