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ad infinitum | A Wisdom Archive on ad infinitum | | ad infinitum A selection of articles related to ad infinitum | |
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ad infinitum
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| ARTICLES RELATED TO ad infinitum | | | |  ad infinitum: Encyclopedia II - Irreducible complexity - Irreducible complexity ICAn early concept of irreducibly complex systems comes from Ludwig von Bertalanffy, a 20th Century Austrian biologist.[3] He believed that complex systems must be examined as complete, irreducible systems in order to understand how they worked. He extended his biological work into a general theory of systems in a book titled General Systems Theory. After James Watson and Francis Crick published the structure of DNA in the early 1950s, GST lost ...
See also:Irreducible complexity, Irreducible complexity - Irreducible complexity IC, Irreducible complexity - Criticism, Irreducible complexity - Definitions, Irreducible complexity - Stated examples, Irreducible complexity - Flagella, Irreducible complexity - Blood clotting cascade, Irreducible complexity - Forerunners, Irreducible complexity - Criticisms of irreducible complexity, Irreducible complexity - Gradual replacement, Irreducible complexity - Handicaps and sexual selection, Irreducible complexity - Falsifiability and experimental evidence, Irreducible complexity - Behe's own Criticisms, Irreducible complexity - God and Irreducible Complexity, Irreducible complexity - Claimed significance Read more here: » Irreducible complexity: Encyclopedia II - Irreducible complexity - Irreducible complexity IC |
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| | | | | ad infinitum: Encyclopedia II - Irreducible complexity - Stated examples Behe and others have suggested a number of biological features that they believe may be irreducibly complex.
Irreducible complexity - Flagella.
The flagella of certain bacteria constitute a molecular motor requiring the interaction of about 40 complex protein parts, and the absence of any one of these proteins causes the flagella to fail to function. Behe holds that the flagellum "engine" is irreducibly complex because if we try to reduce its complexity by positing an earlier and simpler stage of its evolu ...
See also:Irreducible complexity, Irreducible complexity - Irreducible complexity IC, Irreducible complexity - Criticism, Irreducible complexity - Definitions, Irreducible complexity - Stated examples, Irreducible complexity - Flagella, Irreducible complexity - Blood clotting cascade, Irreducible complexity - Forerunners, Irreducible complexity - Criticisms of irreducible complexity, Irreducible complexity - Gradual replacement, Irreducible complexity - Handicaps and sexual selection, Irreducible complexity - Falsifiability and experimental evidence, Irreducible complexity - Behe's own Criticisms, Irreducible complexity - God and Irreducible Complexity, Irreducible complexity - Claimed significance Read more here: » Irreducible complexity: Encyclopedia II - Irreducible complexity - Stated examples |
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| | | | | ad infinitum: Encyclopedia II - Irreducible complexity - Forerunners The argument from irreducible complexity is a descendant of the teleological argument for God (the argument from design or argument from complexity). This states that because certain things in nature are very complicated, they must have been designed, just as the existence of a watch implies the existence of a watchmaker (in William Paley's famous argument of 1802). This argument has a long history and can be traced back at least as far as Cicero's De natura deorum, ii. ...
See also:Irreducible complexity, Irreducible complexity - Irreducible complexity IC, Irreducible complexity - Criticism, Irreducible complexity - Definitions, Irreducible complexity - Stated examples, Irreducible complexity - Flagella, Irreducible complexity - Blood clotting cascade, Irreducible complexity - Forerunners, Irreducible complexity - Criticisms of irreducible complexity, Irreducible complexity - Gradual replacement, Irreducible complexity - Handicaps and sexual selection, Irreducible complexity - Falsifiability and experimental evidence, Irreducible complexity - Behe's own Criticisms, Irreducible complexity - God and Irreducible Complexity, Irreducible complexity - Claimed significance Read more here: » Irreducible complexity: Encyclopedia II - Irreducible complexity - Forerunners |
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| | | | | ad infinitum: Encyclopedia II - Irreducible complexity - Claimed significance Behe argues that organs and biological features which are irreducibly complex cannot be wholly explained by current models of evolution. He argues that:
An irreducibly complex system cannot be produced directly (that is, by continuously improving the initial function, which continues to work by the same mechanism) by slight, successive modifications of a precursor system, because any precursor to an irreducibly complex system th ...
See also:Irreducible complexity, Irreducible complexity - Irreducible complexity IC, Irreducible complexity - Criticism, Irreducible complexity - Definitions, Irreducible complexity - Stated examples, Irreducible complexity - Flagella, Irreducible complexity - Blood clotting cascade, Irreducible complexity - Forerunners, Irreducible complexity - Criticisms of irreducible complexity, Irreducible complexity - Gradual replacement, Irreducible complexity - Handicaps and sexual selection, Irreducible complexity - Falsifiability and experimental evidence, Irreducible complexity - Behe's own Criticisms, Irreducible complexity - God and Irreducible Complexity, Irreducible complexity - Claimed significance Read more here: » Irreducible complexity: Encyclopedia II - Irreducible complexity - Claimed significance |
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| | | | | ad infinitum: Encyclopedia II - Gödel's incompleteness theorem - Proof sketch for the second theorem Let p stand for the undecidable sentence constructed above, and let's assume that the consistency of the system can be proven from within the system itself. We have seen above that if the system is consistent, then p is not provable. The proof of this implication can be formalized in the system itself, and therefore the statement "p is not provable", or "not P(p)" can be proven in the system.
But this last statement is equivalent to p itself (and this equivalence can be proven in the system), so p can ...
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 second theorem |
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| | | | | ad infinitum: 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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| | | | | ad infinitum: 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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| | | | | ad infinitum: 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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| | | | | ad infinitum: 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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| | | | | ad infinitum: 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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| | | | | ad infinitum: Encyclopedia II - Gödel's incompleteness theorem - Gentzen's theorem In 1936 Gerhard Gentzen proved the consistency of first order arithmetic. In itself, the result is rather trivial, since the consistency of first order arithmetic has a very easy proof: the axioms are true—in a mathematically defined sense—the rules of predicate calculus preserve truth and no contradiction is true, hence no contradiction follows from the axioms of first order arithmetic. What makes Gentzen's proof interesting is that it shows much more than merely that first order arithmetic is consistent. Gentzen showed that the consist ...
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 - Gentzen's theorem |
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| | | | | ad infinitum: 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 - 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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| | | | | | ad infinitum: Encyclopedia II - Gödel's incompleteness theorem - First incompleteness theorem Gödel's first incompleteness theorem is perhaps the most celebrated result in mathematical logic. It basically says that:
For any consistent formal theory including basic arithmetical truths, it is possible to construct an arithmetical statement that is true 1 but not included in the theory. That is, any consistent theory of a certain expressive strength is incomplete.
Here, "theory" has the special sense of a set of statements closed under logical inference ru ...
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 - First incompleteness theorem |
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| | | | | ad infinitum: Encyclopedia II - Gödel's incompleteness theorem - Gentzen's theorem In 1936 Gerhard Gentzen proved the consistency of first order arithmetic. In itself, the result is rather trivial, since the consistency of first order arithmetic has a very easy proof: the axioms are true—in a mathematically defined sense—the rules of predicate calculus preserve truth and no contradiction is true, hence no contradiction follows from the axioms of first order arithmetic. What makes Gentzen's proof interesting is that it shows much more than merely that first order arithmetic is consistent. Gentzen showed that the consist ...
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 - Gentzen's theorem |
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| | | | | | | ad infinitum: Encyclopedia II - Karma - Analogs of Karma - God the judge If we accept that the basic ethical purpose of Karma is to behave responsibly, and that the tenet of Karma may be simply stated 'if you do good things, good things will happen to you - if you do bad things, bad things will happen to you', then it is possible for us to identify analogs with other religions that do not rely on Karma as a metaphysical assertion or doctrine.
Karma does not specifically concern itself with salvation - it is just as important within a basic socio-ethical stance. However, as a mechanic, Karma can be identifi ...
See also:Karma, Karma - Karma in the Dharma-based religions, Karma - Hinduism, Karma - Buddhism, Karma - Analogs of Karma - God the judge, Karma - Western interpretation, Karma - New Age and Theosophy, Karma - Psychology Read more here: » Karma: Encyclopedia II - Karma - Analogs of Karma - God the judge |
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| | | | | ad infinitum: Encyclopedia II - Karma - Western interpretation An academic and religious definition was mentioned above. Millions of people believe in it and is a part of many cultures and the psyches of millions of people. Others without religious backgrounds, especially in western cultures or with Christian upbringings, become convinced of the existence of Karma. For some, karma is a more reasonable concept than eternal damnation for the wicked. Spirituality or a belief that virtue is rewarded and sin creates ...
See also:Karma, Karma - Karma in the Dharma-based religions, Karma - Hinduism, Karma - Buddhism, Karma - Analogs of Karma - God the judge, Karma - Western interpretation, Karma - New Age and Theosophy, Karma - Psychology Read more here: » Karma: Encyclopedia II - Karma - Western interpretation |
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| | | | | ad infinitum: Encyclopedia II - J. Allen Hynek - Early Life and Career Hynek was born in Chicago to Czechoslovakian parents. In 1931, Dr. Hynek received a B.S. from the University of Chicago. In 1935, he completed his Ph.D. in astrophysics at Yerkes Observatory. He joined the Department of Physics and Astronomy at Ohio State University in 1936. He specialised in the study of stellar evolution, and in the identification of spectroscopic binaries.
Durring World War II, Hynek was a civilian scientist at the Johns Hopkins Applied Science Laboratory, where he helped to develop the navy's radio proximity fuze.
After the war, Hynek returned to the Department of Physics and As ...
See also:J. Allen Hynek, J. Allen Hynek - Early Life and Career, J. Allen Hynek - Project Grudge and Project Blue Book, J. Allen Hynek - Final years, J. Allen Hynek - UFO books Read more here: » J. Allen Hynek: Encyclopedia II - J. Allen Hynek - Early Life and Career |
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| | | | | ad infinitum: 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 - 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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| | | | | ad infinitum: 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 - 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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