intuitionistic type theory

Type has historically had the following uses: In biology, a type is the specimen or specimens upon which an original species description is based. In computer science, a datatype (sometimes referred to as a 'type') is a collection of values used for computation. In printing, type refers to the metal forms of the letters used in typesetting. In sociology, the terms normal type and ideal type are used. In datin

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Anders Martin-Löf is a Swedish mathematician. He is the brother of Per Martin-Löf. (Who was responsible for a pioneering definition of randomness, as well as a foundation for constructive mathematics based on intuitionistic type theory.) Anders is a is a professor (in mathematical statistics) at the Department of Mathematics of Stockholm University. (His brother Per is also a professor at Stockholm, with joint appointments in the departments of Mathematics and Philosophy. They share an interest in statistics, and in statistic ...

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• Anders Martin-Löf - External link

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In mathematics, the axiom of choice, or AC, is an axiom of set theory. It was formulated in 1904 by Ernst Zermelo. While it was originally controversial, it is now used without embarrassment by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, that either reject the axiom of choice, or even investigate consequences of axioms inconsistent with AC. Intuitively speaking, AC says that given a collection of bins, each containing at least one object, then exactly one ob ...

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• Axiom of choice - Statement
• Axiom of choice - Usage
• Axiom of choice - Independence of AC
• Axiom of choice - Weaker forms of AC
• Axiom of choice - Results requiring AC or weaker forms
• Axiom of choice - Results requiring ¬AC
• Axiom of choice - Results requiring choice in intuitionistic logic though not classically
• Axiom of choice - Quotes

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Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X." In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo. Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other ...

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Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

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Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventionally the story of modern proof theory is seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's problem of reducing all mathematics to a finitist formal system, then his incompleteness theorems showed that was unattainable. All of this work was carried out with the pr ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is not assumed, the assumption of the axiom of choice is sufficient to obtain the law of excluded middle as a theorem. To see this, for any proposition let be the set and let be the set (see Set-builder notation). By the axiom of choice, there will exist a choice function for the set (note that, although the axiom of choice isn't classically required in order to obtain choice functions for finite sets, it ...

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Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic, though not classically, Axiom of choice - Quotes

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Main article: Tableau systems Tableau systems apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Main article: Structural proof theory Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz; the definition is slightly more complex, we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Gira ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Main article: Ordinal analysis Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Main article: Consistency proof As we have discussed, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable Pi-0-1 sentences) are finitarily true; once so grounded w ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to prove the negation of some standard facts. Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true. There exists a model of ZF¬C in which there is a function f from the real numbers to the real nu ...

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Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

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The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X. Stated more formally: Let X be a set of non-empty sets. Then there exists a choice function f defined on X. In other words, there exists a function f defined on X, such that for each set s in X, f(s) is an element of s. Another formulation of the axiom of choice states: Given any set of mutually disjoint non-empty sets, there exists at least one set that contains exactly one eleme ...

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Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

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By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (ZF). This means that neither it nor its negation can be proven to be true in ZF. Consequently, assuming the axiom of choice, or its negation, will never lead to a contradiction that could not be obtained without that assumption. So the decision whether or not it is appropriate to make use of the axiom of choice in a proof cannot be made by appeal to other axioms of s ...

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Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Independence of AC

Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is not assumed, the assumption of the axiom of choice is sufficient to obtain the law of excluded middle as a theorem. To see this, for any proposition let be the set and let be the set (see Set-builder notation). By the axiom of choice, there will exist a choice function for the set (note that, although the axiom of choice isn't classically required in order to obtain choice functions for finite sets, it ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring choice in intuitionistic logic though not classically

However, the proofs used in everyday mathematical practice are almost never like the formal proofs in proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, and given enough time and patience. For most mathematicians, writing a fully formal proof would have all the drawbacks of programming in machine code. Formal proofs are constructed, with the help of computers, in automated theorem proving. Significantly, these proofs can be checked automatically by com ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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The three most well known proof calculi are: The Hilbert-style calculi The natural deduction calculus The sequent calculus To say these are proof calculi, rather than proof systems, is to say they are flexible frameworks for the study of many kinds of logical consequence relations. Each of these can formalise propositional or predicate logics of either the classical or intuitionistic flavour, or almost any modal logic studied, many substructural logics, such as relevance logic or lin ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

Read more here: » Proof theory: Encyclopedia II - Proof theory - Kinds of proof calculus