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Gerhard Gentzen | A Wisdom Archive on Gerhard Gentzen | | Gerhard Gentzen A selection of articles related to Gerhard Gentzen | |
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Gerhard Gentzen
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ARTICLES RELATED TO Gerhard Gentzen | |
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| | |  Gerhard Gentzen: Encyclopedia II - Peano axioms - Categorical interpretationThe Peano axioms may be interpreted in the general context of category theory. Let US1 be the category of pointed unary systems; i.e. US1 is the following category:
The objects of US1 are all ordered triples (X, x, f), where X is a set, x is an element of X, and f is a set map from X to itself.
For each (X, x, f), (Y, y, g) in US1, HomUS1((X, x, f), ...
See also:Peano axioms, Peano axioms - The axioms, Peano axioms - Peano arithmetic, Peano axioms - Existence and uniqueness, Peano axioms - Binary operations and ordering, Peano axioms - Categorical interpretation, Peano axioms - Metamathematical discussion Read more here: » Peano axioms: Encyclopedia II - Peano axioms - Categorical interpretation |
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| | | | Gerhard Gentzen: Encyclopedia II - Proof theory - History 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 ...
See also: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 - History |
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| | | | Gerhard Gentzen: Encyclopedia II - Logic - Nature of logic Because of its fundamental role in philosophy, the nature of logic has been the object of intense dispute: it is not possible clearly to delineate the bounds of logic in terms acceptable to all rival viewpoints. Despite that controversy, the study of logic has been very coherent and technically grounded. In this article, we first characterise logic by introducing fundamental ideas about form, then by outlining some schools of thought, as well as by giving a brief overview of logic's history, an account of its relationship to other sciences, and finally, an exposition of some of logic's essential concepts.
Logic - I ...
See also:Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical? Read more here: » Logic: Encyclopedia II - Logic - Nature of logic |
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| | | | Gerhard Gentzen: Encyclopedia II - Sequent - Intuitive meaning The intuitive meaning of a sequent such as the one given above is that under the assumption of Γ the conclusion of Σ is provable. In a classical setting, the formulae on the left of the turnstile are interpreted conjunctively while the formulae on the right are considered as a disjunction. This means that, when all formulae in Γ hold, then at least one formula in Σ also has to be true. If the succedent is empty, this is interpreted as falsity, i.e. means that Γ proves falsity and is thus inconsistent. On the other hand an empty anteced ...
See also:Sequent, Sequent - Explanation, Sequent - Intuitive meaning, Sequent - Example, Sequent - Property, Sequent - Rules, Sequent - Variations, Sequent - History Read more here: » Sequent: Encyclopedia II - Sequent - Intuitive meaning |
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| | | | Gerhard Gentzen: Encyclopedia II - Quantification - Notation for quantifiers The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for the word "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for the word "exists". Correspondingly, quantified expressions are constructed as follows,
where "P" denotes a formula. Many variant notations are used, such as
All of these variations apply to univers ...
See also:Quantification, Quantification - Quantification in natural language, Quantification - Need for quantifiers in mathematical assertions, Quantification - Nesting of quantifiers, Quantification - Range of quantification, Quantification - Notation for quantifiers, Quantification - Formal semantics, Quantification - Paucal multal and other degree quantifiers, Quantification - History of formalisation Read more here: » Quantification: Encyclopedia II - Quantification - Notation for quantifiers |
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| | | | Gerhard Gentzen: Encyclopedia II - Gödel's incompleteness theorem - Second incompleteness theorem Gödel's second incompleteness theorem can be stated as follows:
For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.
(Proof of the "if" part:) If T is inconsistent then anything can be proved, including that T is consistent. (Proof of the "only if" part:) If T is consistent then T does not i ...
See also:Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Second incompleteness theorem |
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| | | | Gerhard Gentzen: Encyclopedia II - List of philosophers - Notes
Note O: - For more information about this person's contribution to philosophy, see his/her entry in The Oxford Companion to Philosophy. Oxford University Press; 1995. ISBN 0198661320
Note R: - For more information about this person's contribution to philosophy, see his/her entry in the Concise Routledge Encyclopedia of Philosophy. Routledge; 2000. ISBN 0415223644
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See also:List of philosophers, List of philosophers - A, List of philosophers - B, List of philosophers - C, List of philosophers - D, List of philosophers - E, List of philosophers - F, List of philosophers - G, List of philosophers - H, List of philosophers - I, List of philosophers - J, List of philosophers - K, List of philosophers - L, List of philosophers - M, List of philosophers - N, List of philosophers - O, List of philosophers - P, List of philosophers - Q, List of philosophers - R, List of philosophers - S, List of philosophers - T, List of philosophers - U, List of philosophers - V, List of philosophers - W, List of philosophers - X, List of philosophers - Y, List of philosophers - Z, List of philosophers - Notes, List of philosophers - General philosophy lists, List of philosophers - General philosophy topics, List of philosophers - General online philosophy resources Read more here: » List of philosophers: Encyclopedia II - List of philosophers - Notes |
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| | | | Gerhard Gentzen: Encyclopedia II - Quantification - Notation for quantifiers The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for the word "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for the word "exists". Correspondingly, quantified expressions are constructed as follows,
where "P" denotes a formula. Many variant notations are used, such as
All of these variations apply to univers ...
See also:Quantification, Quantification - Quantification in natural language, Quantification - Need for quantifiers in mathematical assertions, Quantification - Nesting of quantifiers, Quantification - Range of quantification, Quantification - Notation for quantifiers, Quantification - Formal semantics, Quantification - Paucal multal and other degree quantifiers, Quantification - History of formalisation, Quantification - Links Read more here: » Quantification: Encyclopedia II - Quantification - Notation for quantifiers |
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| | | | Gerhard Gentzen: Encyclopedia II - Logic - Nature of logic Because of its fundamental role in philosophy, the nature of logic has been the object of intense disputation; and it is not possible to give a clear delineation of the bounds of logic in terms acceptable to all rival viewpoints. Nonetheless, the study of logic has, despite this controversy, been very coherent and technically grounded. Here we characterise logic, first by introducing the fundamental ideas about form and then by outlining some of the different schools of thought as well as giving a brief overview of its history, an account of its relationship to other sciences, and--finally--an expositi ...
See also:Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical? Read more here: » Logic: Encyclopedia II - Logic - Nature of logic |
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| | | | Gerhard Gentzen: Encyclopedia II - 1909 - Events
1909 - January – March.
January 16 - Ernest Shackleton's expedition finds the magnetic South Pole.
January 28 - United States troops leave Cuba after being there since the Spanish-American War.
February 12 - The National Association for the Advancement of Colored People (NAACP) is founded.
February 23 - The Silver Dart makes the first powered flight in Canada and the British Empire.
February 24 - The Hudson Motor Car Company is founded.
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See also:1909, 1909 - Events, 1909 - January – March, 1909 - April – June, 1909 - July – September, 1909 - October – December, 1909 - Month/date unknown, 1909 - Births, 1909 - January, 1909 - February, 1909 - March, 1909 - April, 1909 - May, 1909 - June, 1909 - July, 1909 - August, 1909 - September, 1909 - October, 1909 - November, 1909 - December, 1909 - Deaths, 1909 - Date unknown, 1909 - Nobel Prizes Read more here: » 1909: Encyclopedia II - 1909 - Events |
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| | | | Gerhard Gentzen: Encyclopedia II - Gödel's incompleteness theorem - Second incompleteness theorem Gödel's second incompleteness theorem can be stated as follows:
For any formal theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.
(Proof of the "if" part:) If T is inconsistent then anything can be proved, including that T is consistent. (Proof of the "only if" part:) If T is consistent then T does not i ...
See also:Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Second incompleteness theorem |
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| | | | Gerhard Gentzen: Encyclopedia II - Gödel's incompleteness theorem - Proof sketch for the first theorem The main problem in fleshing out the above mentioned proof idea is the following: in order to construct a statement p that is equivalent to "p cannot be proved", p would have to somehow contain a reference to p, which could easily give rise to an infinite regress. Gödel's ingenious trick, which was later used by Alan Turing to show that the Entscheidungsproblem is unsolvable, will be described below.
To begin with, every formula or statement that can be formulated in our system gets a unique number, called ...
See also:Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Proof sketch for the first theorem |
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| | | | Gerhard Gentzen: Encyclopedia II - Gödel's incompleteness theorem - Minds and machines Many scholars have debated over what Gödel's incompleteness theorem implies about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.
One of the earliest attempts to use incompleteness to reason about human intelligence was by Gödel himself in his 1951 Gibbs lecture entitled "Some basic theorems on the foundations of mat ...
See also:Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Minds and machines |
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| | | | Gerhard Gentzen: Encyclopedia II - Gödel's incompleteness theorem - Discussion and implications The incompleteness results affect the philosophy of mathematics, particularly viewpoints like formalism, which uses formal logic to define its principles. One can paraphrase the first theorem as saying that "we can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods."
On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical "truth" and "falsehood" are ...
See also:Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Discussion and implications |
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| | | | Gerhard Gentzen: Encyclopedia II - Gödel's incompleteness theorem - Examples of undecidable statements It should be noted that there are two distinct senses of the word "undecidable" in use. The first of these is the sense used in relation to Gödel's theorems, i.e., that of a statement being neither provable nor refutable, in some specified deductive system. The second sense is used in relation to recursion theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes/no answer. Such a problem is said to be undecidable if there is no recursive function that correctly answer ...
See also:Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Examples of undecidable statements |
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| | | | Gerhard Gentzen: Encyclopedia II - Gödel's incompleteness theorem - Meaning of Gödel's theorems Gödel's theorems are theorems in first-order logic, and must ultimately be understood in that context. In formal logic, both mathematical statements and proofs are written in a symbolic language, one where we can mechanically check the validity of proofs so that there can be no doubt that a theorem follows from our starting list of axioms. In theory, such a proof can be checked by a computer, and in fact there are computer programs that will check the validity of proofs. (Automatic proof verification is closely related to automated theo ...
See also:Gödel's incompleteness theorem, Gödel's incompleteness theorem - First incompleteness theorem, Gödel's incompleteness theorem - Gödel's Theorem, Gödel's incompleteness theorem - Second incompleteness theorem, Gödel's incompleteness theorem - Gentzen's theorem, Gödel's incompleteness theorem - Meaning of Gödel's theorems, Gödel's incompleteness theorem - Examples of undecidable statements, Gödel's incompleteness theorem - Misconceptions about Gödel's theorems, Gödel's incompleteness theorem - Discussion and implications, Gödel's incompleteness theorem - Minds and machines, Gödel's incompleteness theorem - Proof sketch for the first theorem, Gödel's incompleteness theorem - Proof sketch for the second theorem, Gödel's incompleteness theorem - Footnotes Read more here: » Gödel's incompleteness theorem: Encyclopedia II - Gödel's incompleteness theorem - Meaning of Gödel's theorems |
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