2250 Lecture Notes - Null Set, Tim Hortons, Reggiane Re.2000
5 Feb 2013
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Document Summary
The language sl has a surprising feature: it is compact. a (possibly infinite) set, , of sentences of sl is t-f inconsistent if and only if at least one finite subset of is t-f inconsistent. If a finite subset, , of a set, , is t-f inconsistent then there is no truth value assignment on which every sentence in is true. Adding more sentences to to create the infinite set, , is not going to change this fact. There is still going to be no truth value assignment that makes all the sentences in true because there is already no truth value assignment that makes just some of them (those in ) true. But it is rather surprising that the converse should be true as well: that if is t-f inconsistent then at least one finite subset of must be t-f inconsistent. There is no problem seeing why that should be the case if is itself finite.
