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In mathematical logic, a sentence σ is called independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
In mathematical logic, a sentence is a formula with no free variables; therefore, a sentence is either true or false in a given structure. ...
In mathematical logic, a theory is usually defined as a set of first-order sentences (closed first-order formulas). ...
Sometimes, σ is said (synonymously) to be undecidable from T; however, this usage risks confusion with the distinct notion of the undecidability of a decision problem. In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ...
Many interesting statements in set theory are independent of Zermelo-Fraenkel set theory(ZF). It is possible for the statement "σ is independent from T" to be itself independent from T. This reflects the fact that statements about proofs of mathematical statements when represented in mathematics become themselves mathematical statements. Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...
Usage note Some authors say that σ is independent of T if T simply cannot prove σ, and do not necessarily assert by this that T cannot refute σ. These authors will sometimes say "σ is independent of and consistent with T" to indicate that T can neither prove nor refute σ.
Theorems relevant to independence Kurt Gödel proved the completeness theorem and the incompleteness theorem Kurt Gödel (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher of mathematics. ...
In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...
In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. ...
The completeness theorem states (Assuming ZFC) A theory T is consistent iff T has a model. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
The incompleteness theorem states (Assuming ZF) In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers, one can construct a statement that can be neither proved nor disproved within that system. Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...
Independence results in set theory The following statements in set theory are known to be independent of ZF, granting that ZF is consistent (see also the list of statements undecidable in ZFC): The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. ...
The following statements (none of which have been proved false) cannot be proved in ZFC to be independent of ZFC, even if the added hypothesis is granted that ZFC is consistent. However, they cannot be proved in ZFC (granting that ZFC is consistent), and few working set theorists expect to find a refutation of them in ZFC. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...
Suslins problem in mathematics is a question about orders posed by M. Suslin in the early 1920s. ...
The following statements are inconsistent with the axiom of choice, and therefore with ZFC. However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF. However ZF cannot prove that they are independent of ZF, even with the added hypothesis that ZF is consistent. In mathematics, a strongly inaccessible cardinal is an uncountable cardinal number κ that is regular and a strong limit cardinal. ...
In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ...
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