Sentence examples for usual proof from inspiring English sources

In fact, classical arguments seem to be necessary for metamathematics for example, in the usual proof of Gödel's completeness theorem even for intuitionistic type theory.

Of course, Gödel's completeness theorem shows that there are plenty of Boolean local topoi to model pure classical type theory in, but the usual proof of their existence requires nonconstructive arguments.

The Osgood proof is acknowledged directly by Carathéodory [2] where the ideas involved in the usual proof of today, via normal families, are presented.

The determination of the constant 2 π in the usual proof of Stirling's formula (7) or Stirling's asymptotic formula Γ ( x ) ∼ 2 π x x − 1 / 2 e − x, x → ∞, (15).

The usual proofs spell things out more explicitly (and in fact, our proof of Proposition 3.12 is extracted from the standard proofs of Theorem 7.10).

A key method in the usual proofs of the first incompleteness theorem is the arithmetization of the formal language, or Gödel numbering: certain natural numbers are assigned to terms, formulas, and proofs of the formal theory F. There are different ways of doing this; one standard approach is sketched here (for a rather different method of coding, see, e.g., Boolos & Jeffrey 1989).

But, as Kripke points out, the usual SQML proof of CBF also depends essentially on an application of Necessitation to an open formula derived by universal instantiation — the same "flaw" that infects the proof of NE. (See the inference from line 1 to line 2 in the supplementary document Proof of the Converse Barcan Formula in S5).

When Daniel Pellegrine decided not to wear his usual bullet-proof vest to the concert he was performing at in Campinas, Sao Paulo, last Saturday night, it proved to be a fatal mistake.

Hence, the usual SQML proofs of all three actualistically unacceptable principles fail in Kripke's system.

And on the subject of compassion, President Bush has more than the usual burden of proof.

The hypotheses about Leos' intestines and Sagittarians' arms were less than 5% likely to have come about by chance, satisfying the usual standards of proof of a relationship.