Definition·D105
Provability
A theory proves a formula when some proof of it from the theory exists.
For a language
,
, and
:
proves
, written
,
In words
A theory proves a formula exactly when some proof of the formula from the theory exists.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 · A09 (computed from the citation graph, not asserted).
Remarks
The syntactic, purely mechanical counterpart of semantic entailment
-
(single turnstile) asks only whether a finite, checkable sequence of formulas exists, never mentioning a structure or an assignment. Soundness will show
(whatever is provable is true in every model); Gödel's completeness theorem will show the converse.