MIT24_241F09_lec13 - Logic I - Session 13 Plan Damien on...

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Unformatted text preview: Logic I - Session 13 Plan Damien on psets Quick summary of completeness Compactness Limitations of SL Intro to PL Completeness P { P } is C-SD { P } a MC-SD set * If * is MC-SD then * is TF-C { P } a TF-C set * { P } is TF-C P Compactness A cool result of completeness: Compactness : is TF-C iff every nite subset of is TF-C. So: a set is TF-IC only if a nite subset of is TF-IC. So, intuitively, theres no TF inconsistency that you need an innite number of SL sentences to get! Lets prove compactness by proving each direction. Compactness First, left-to-right: If is TF-C, then every nite subset of is TF-C. If there were a subset - such that no TVA m.e.m. - true, then there would be no TVA m.e.m. true. Now, right-to-left: If every nite subset of is TF-C, then is TF-C....
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MIT24_241F09_lec13 - Logic I - Session 13 Plan Damien on...

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