Let A be set of closed sentences ( informally " axioms " ) and \ langle A \ rangle the set of closed sentences provable from A under a meta-theoretical deductive system such as informal mathematics.
42.
Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based.
43.
Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based.
44.
A "'syntactically consistent theory "'is a theory from which not every sentence in the underlying language can be proven ( with respect to some deductive system which is usually clear from context ).
45.
When reasoning about the meta-theoretic properties of a deductive system in a proof assistant, it is sometimes desirable to limit oneself to first-order representations and to have the ability to ( re ) name assumptions.
46.
There is thus, on the one hand, the notion of " completeness of a deductive system " and on the other hand that of " completeness of a set of non-logical axioms ".
47.
G�del's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language.
48.
On the other hand, he draws continually on examples from common experience, and his semiotic is not contained in a mathematical or deductive system and does not proceed chiefly by drawing necessary conclusions about purely hypothetical objects or cases.
49.
Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.
50.
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means.
