User:IssaRice/Computability and logic/Gödel's completeness theorem: Difference between revisions
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[[../Semantic completeness|semantic completeness]] | [[../Semantic completeness|semantic completeness]] | ||
==Different formulations== | |||
There are two equivalent formulations of the completeness theorem. One phrases the theorem in terms of "logical consequence implies provable" and the other phrases it in terms of "consistent implies satisfiable". | |||
# (for all sets of sentences <math>\Gamma</math> and all sentences <math>\phi</math>?) If <math>\Gamma \models \phi</math>, then <math>\Gamma \vdash \phi</math> | |||
# "The completeness theorem says that, if a bunch of sentences is consistent in the sense of not entailing a contradiction [in a standard system of first-order logic] then it has a model."<ref>https://math.stackexchange.com/a/436257/35525</ref> See also Enderton.<ref>Enderton. A Mathematical Introduction to Logic. p. 135.</ref> | |||
==References== | ==References== | ||
<references/> | <references/> | ||
Revision as of 07:12, 21 December 2018
Different formulations
There are two equivalent formulations of the completeness theorem. One phrases the theorem in terms of "logical consequence implies provable" and the other phrases it in terms of "consistent implies satisfiable".
- (for all sets of sentences and all sentences ?) If , then
- "The completeness theorem says that, if a bunch of sentences is consistent in the sense of not entailing a contradiction [in a standard system of first-order logic] then it has a model."[1] See also Enderton.[2]
References
- ↑ https://math.stackexchange.com/a/436257/35525
- ↑ Enderton. A Mathematical Introduction to Logic. p. 135.