불완전성 정리

노트

위키데이터

• ID : Q200787

말뭉치

1. Giving a mathematically precise statement of Godel's Incompleteness Theorem would only obscure its important intuitive content from almost anyone who is not a specialist in mathematical logic.
2. It was even more shocking to the mathematical world in 1931, when Godel unveiled his incompleteness theorem.
3. But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream.
4. That’s Gödel’s first incompleteness theorem.
5. Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic.
6. Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems.
7. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem.
8. The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers.
9. We will argue that Gödel’s completeness theorem is a kind of completability theorem, and Gödel-Rosser’s incompleteness theorem is a kind of incompletability theorem in a constructive manner.
10. But demonstrating an re-incompletable theory is a difficult task and it is in fact Gödel’s Incompleteness Theorem.
11. Let us note that Gödel’s original first incompleteness theorem showed the existence of some theory which was not soundly re-completable.
12. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent).
13. The present entry surveys the two incompleteness theorems and various issues surrounding them.
14. Gödel established two different though related incompleteness theorems, usually called the first incompleteness theorem and the second incompleteness theorem.
15. This is, however, incorrect, for the incompleteness theorem does not deal with provability in any absolute sense, but only concerns derivability in some particular formal system or another.
16. We give some information about new proofs of the incompleteness theorems, found in 1990s.
17. The conclusion that follows, that the consistency of arithmetic cannot be proved within arithmetic, is known as Gödel’s second incompleteness theorem.
18. In Section 8.2, we shall discuss some general properties of recursive logics, yielding a proof of Gödel’s Incompleteness Theorem.
19. Gödel's second incompleteness theorem is obtained by formalizing the proof of Gödel's first incompleteness theorem.
20. For an account of how the Incompleteness Theorems have been misunderstood and mishandled see Torkel Franzen's Gödel's Incompleteness Theorem: an incomplete guide to its use and abuse.
21. I recommend both Peter Smith's Gödel's Theorems and Raymond Smullyan's Gödel's Incompleteness Theorems.
22. It is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic.
23. We generalize Gödel’s incompleteness theorems for arithmetically definable theories.
24. In logic, an incompleteness theorem expresses limitations on provability within a (consistent) formal theory.
25. To some extent, Gödel’s incompleteness theorems have always had an air of mystery about them, or at least a reputation of being exceedingly difficult or subtle.
26. This lecture, given at Beijing University in 1984, presents a remarkable (previously unpublished) proof of the Gödel Incompleteness Theorem due to Kripke.
27. Gödel's first incompleteness theorem states that all consistent axiomatic formulations of number theory which include Peano arithmetic include undecidable propositions (Hofstadter 1989).
28. Furthermore, I demonstrate that this answer has interesting connections to the halting problem from computability theory and to Gödel's incompleteness theorem from mathematical logic.
29. Godel's incompleteness theorem Does Godel's Theorem really prove arithmetic systems incomplete as a stand-alone system?
30. Kurt Godel proved (using meta-mathematics) in 1930 his so-called "incompleteness theorem.
31. Yet this conclusion followed from Gödel's next results, his incompleteness theorems, which had consequences for both first-order and second-order logic.
32. She even gets the year in which Gödel first proved his incompleteness theorem wrong—it was 1930, not 1929 as she suggests on page 156.
33. Yourgrau discusses Gödel's incompleteness theorems more adequately than Goldstein does.
34. Those who want a clear, simplified, but fundamentally correct explanation of the incompleteness theorems should read Ernst Nagel and James R. Newman's short book Gödel's Proof.
35. In 1931 a German mathematician named Kurt Gödel published a paper which included a theorem which was to become known as his Incompleteness Theorem.
36. Gödel proved his Incompleteness Theorem in a rather bizarre but effective manner.
37. In the same light, Gödel's Incompleteness Theorem is a theorem of HLS, so it is called true.
38. What all this boils down to is that Gödel's Incompleteness Theorem apples equally to machines and minds.
39. Gödel’s second incompleteness theorem gives a specific example of such an unprovable statement.
40. Gödel’s Incompleteness Theorem applies not just to math, but to everything that is subject to the laws of logic.
41. So according to Gödel’s Incompleteness theorem, the Infidels cannot be correct.
42. Gödel’s Incompleteness Theorem definitively proves that science can never fill its own gaps.

메타데이터

위키데이터

• ID : Q200787

Spacy 패턴 목록

• [{'LOWER': 'gödel'}, {'LOWER': "'s"}, {'LOWER': 'incompleteness'}, {'LEMMA': 'theorem'}]