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Gödel
Gödel (ˈg{obar}ːdəl) The name of Kurt Gödel (born 1906), Austrian mathematician, used attrib. and in the possessive to designate his metamathematical theorems and related techniques and constituents; as Gödel number, Gödel numbering; Gödel's proof; Gödel('s) theorem, the demonstration (first publish... Oxford English Dictionary
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Gödel number for contradicting modus ponens? When Gödel numbered statements, for instance modus ponens and connectives got their own numbers, does it matter which number each connective gets as long as they are differ...
For Gödel original numbering, see: * Jean van Heijenoort (editor), From Frege to Gödel: A Source Book in Mathematical Logic (1967), page 600: The
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Is there a non-contradictory non-trivial axiomatic system in which Gödel's theorem is undecidable? Gödel's Incompleteness Theorem states that in any non-contradictory non-trivial axiomatic system there are certain sta...
Robinson arithmetic is non-trivial enough that the incompleteness theorem applies to it, but as far as I know not strong enough to prove the incompleteness theorem itself.
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Gödel's Incompleteness Theorem - Diagonal Lemma In proving Gödel's incompleteness theorem, why does he needed the Diagonal Lemma or the Fixed Point Theorem for building a formula $\phi$ that spoke about itself? Can't ...
If I understand your notation correctly, the formula (whose Gödel number is) $j$ does not assert that $j$ itself has no free variables, merely that $\psi
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Gödel's proof method and fundamental theorem of arithmetic I am a novice to Gödel's proof (the theorem that consistency contradicts completeness), and, as it seems to me, he uses the fundamental theorem of arithmetic ...
Note that $\sf PA$ proves the fundamental theorem of arithmetic (the usual proof goes through just fine), then $\sf PA$ proves that the encoding is unique. It is important to note that the function mapping a formula to its number is _not_ internal to the model of $\sf PA$, a model of $\sf PA$ is not...
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Is a Gödel sentence logically valid? This might be an elementary question, but I am just beginning to learn logic theory. From wikipedia article on Gödel's incompleteness theorems > Any effectively generated theory...
No, a Gödel sentence is not logically valid. of $T$ in which the Gödel sentence is false.
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Do Gödel numbers have a practical use? Is there any example of Gödel numbers being actually used in practice? If so for what purpose?
The Information Age is based on the Gödel number. ), and you tell the computer to execute the program with that Gödel number.
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Gödel's Paradox --- Every set of formulas is consistent I am sure I have made a gross misunderstanding of Gödel's completeness theorems, as to me, it seems to follow that all sets of formulas are consistent. Let $\Ga...
Two things. First of all, $\Gamma \vdash \psi$ implying $\Gamma \models \psi$ is known as the _Soundness Theorem_ for the proof system $\vdash$ (i.e., "true premises do not prove false conclusions"). Now on your purported proof, the flaw occurs at "Then $\Gamma \not\models (\neg \psi)$". Namely, $\G...
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Does Gödel's incompleteness theorem invoke a Law of Excluded Middle contradiction? Does Gödel's incompleteness theorem cause the Law of Non Contradiction to contradict its self? If so, would this be a considered a co...
Gödel's theorem does not appeal to notion of "truth" of a conjecture _per se_ \- merely to its provability within a given set of axioms.
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Can two distinct formulae (or series of formulae) have the same Gödel number? As I am studying Gödel's incompleteness theorem I am wondering if two distinct formulae or series of formulae can have the same Gödel numbe...
guaranteeing by the fundamental theorem of arithmetic: there is only one decomposition of a number in product of primes, and that is what is used by Gödel
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Modern book on Gödel's incompleteness theorems in all technical details Is there a modern book on Gödel's incompleteness theorems that goes into each and every technical aspect of the proof of them (a classical one, i...
incompleteness theorems and their related computability and philosophical concepts, the best modern reference is Peter Smith's book _An Introduction to Gödel's
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How do I best learn Gödel's incompleteness theorem? I'm a math beginner. I am a self-learner. A while ago, I finished reading “How to Prove It: A Structured Approach, 2nd Edition” by Daniel J. Velleman. Now, I want t...
You can see Gödel's Incompleteness Theorems. Two good books are : * Torkel Franzén, Gödel's theorem : An incomplete guide to its use and abuse (2005) * Peter Smith, An Introduction to Gödel's
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Infinitely many nonequivalent unprovable statements in ZFC because of Gödel's incompleteness theorem? Am I right in thinking that there are countably infinitely many nonequivalent unprovable statements in ZFC due to t...
First of all, "countably" is redundant, because there are only countably many sentences in the first place. With that out of the way, you are correct. Assume there are only finitely many unprovable statements (up to equivalence modulo ZFC) $\sigma_1 \ldots \sigma_n$, and that ZFC is consistent. Then...
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Help understanding Gödel's theorems? What are the prerequisites to even begin to understand Gödel's theorems? I'm reading Hofstadter's book but would like a more fundamental approach to understanding these theorems. I...
There is a book on sale at Google Play called _An Introduction to Gödel's Theorems_. You can also buy _Gödel's Theorem: An Incomplete Guide to its Use and Abuse_.
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Is ZFC ω-consistent over ZF? Gödel proved that if ZF is consistent, then ZFC is also consistent. He did this by showing that his constructible universe is a model of ZFC. Gödel also introduced the notion of an $\omega...
Any witness of $\omega$-inconsistency of ZFC can be effectively transformed to a witness of $\omega$-inconsistency of ZF as follows. Suppose $\phi(x)$ is a formula such that for each numeral $n$, ZFC proves $\phi(n)$ and also $(\exists x \in \omega)\neg \phi(x)$. Let $\psi(x) = \phi^L(x)$ be obtaine...
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