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The quality of being “true” is dependent on the model one is using. One can not talk about “truth” without being in a model. (Assuming we are talking about st
by skh 8y ago
The quality of being “true” is dependent on the model one is using. One can not talk about “truth” without being in a model. (Assuming we are talking about standard mathematical logic.). A statement in a first order system is provable if and only if it is true in all models for that system.
- joe_the_user 8y agoThat's it, plus a model is an assignment of a truth value to every well-formed statement of logic language.
- eximius 8y agoYou either don't understand the point of the theorems or are being contrarian. In the context for the Incompleteness theorems, there are two 'kinds' of truths: a more informal kind used by all of us everyday and the mathematical kind as in, proven true under a given system. The entire purpose of the theorems is to establish that there exists theorems in the first set that are not in the second set, while being expressable in the system. To deny the existence of the first kind of true statements ignores the entire purpose of it all.
- skh 8y agoI used the term ‘truth’ as used in mathematical logic. In a given model a statement can be true or false. Under a given axiomatic system a statement is either provable or not. We don’t use the word “true” when dealing with statement under an axiomatic system. We do use the word when dealing with a statement in a given model. Your third paragraph doens’t make sense. In first order logic a theorem is a statement that is true in all models and is one that is provable. This is a result of the Completeness Theorem.