Sentence examples for incompleteness results from inspiring English sources

This will make easy to justify deep and astonishing (counter-intuitive) incompleteness results about computers and similar machines.

This result and others notably mathematician-logician Kurt Gödel's incompleteness results—dashed the hopes, held by some mathematicians, of discovering a formal system that would reduce the whole of mathematics to methods that (human) computers could carry out.

"Not only can this procedure be repeated any finite number of times, it can also be iterated into the transfinite". Feferman investigated how far into the transfinite such progressions could be carried, and established fundamental completeness and incompleteness results for them, depending on the way that they are generated; these results made essential use of his work on arithmetization.

He obtained abstract versions of incompleteness results apparently already in 1922.

However, no one in Hilbert's school realized the extent of the difficulty until 1930, when Gödel announced his incompleteness results.

Gödel himself developed an argument against the conventionalist philosophy of mathematics of logical positivism, and of Carnap's in particular, based on the incompleteness results (Gödel 1953/9).

Because the FARS-based registry's incompleteness resulted primarily from incomplete ascertainment of wheelchair use not missing crash episodes almost all unmatched cases could be identified in FARS as pedestrian crashes that failed to code the pedestrian as using a wheelchair (with 252 of the total of 255 identified cases ultimately identifiable in FARS).

This also easily yields a weak version of the incompleteness result: the set of sentences provable in arithmetic can be defined in the language of arithmetic, but the set of true arithmetical sentences cannot; therefore the two cannot coincide.

One might at first expect this limitation to be resolvable by the inclusion of additional axioms, but Gödel showed that the incompleteness result still holds when first-order arithmetic is extended with an arbitrary finite set of axiom schemas (or, more generally, an arbitrary recursive set of axioms).

One should not get confused here: "Gödel's theorem" is the general incompleteness result of Gödel which concerns a large class of formal systems, while the "Gödel sentence" is the constructed, formally undecidable sentence which varies from one formal system to another.

In this situation, incompleteness resulted from not knowing participants' longer-term outcomes beyond the last 24-month interview.