cartesian closed categories

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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Let C be a category with binary products and let Y and Z be objects of C. The exponential object ZY can be defined as a universal morphism from the functor –×Y to Z. (The functor –×Y from C to C maps objects X to X×Y and morphisms φ to φ×idY). Explicitly, the definition is as follows. An object ZY, together with a morphism is an exponential object if for any object X and morphism g : (X×Y) → ZSee also:

Exponential object, Exponential object - Definition, Exponential object - Examples

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In the category of sets, the exponential object ZY is the set of all functions from Y to Z. The map is just the evaluation map which sends the pair (f, y) to f(y). For any map the map is the curried form of g: In the category of topological spaces, the exponential object ZY exists provided that Y< ...

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Exponential object, Exponential object - Definition, Exponential object - Examples

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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: 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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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

Read more here: » Proof theory: Encyclopedia II - Proof theory - Formal and informal proof

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

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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

Read more here: » Proof theory: Encyclopedia II - Proof theory - Consistency proofs