Completeness

For the sake of completeness, let us mention that the ring R, considered as a module over itself, has submodules of arbitrarily large finite length.

His childish idea was, in fact, a pushing to the extremity of mathematical precision what is everywhere known as Grimm's Law—an aggrandizement of rough rules to ideal completeness.

In a like case the modern scholar encounters an even greater danger, because in the detailed investigation of manifold subjects, he runs the risk of scattering his energies and of losing himself in disconnected knowledge, without supplementing the incomplete, as the ancients succeeded in doing, by the completeness of his own personality.

THEOREM 37°. (Gödel's completeness theorem 1930.) In the predicate calculus H: (a) If #92;vDashF [or even if #92;aleph#95;0-#92;vDashF], then #92;vdashF. If E#95;1,...,E#95;k#92;vDashF [or even if E#95;1,...,E#95;k#92;#92;aleph#95;0-#92;vDashF], then E#95;1,...,E#95;k#92;vdashF. (b) […]