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It is an interesting question that whether some mathematical statement(s) can or cannot prove some other statement. To show a statement is provable, simply present a proof. But is there a way to show a statement is unprovable? For example, the Euclid's fifth postulate turned out to be unprovable from other postulates. Because one can find a geometry where all Euclid's postulates are ture (including the fifth) and another where all but the fifth are true. Since proof preserves truth one can say the fifth is unprovable. I am going to demonstrate a simple technique of showing the three axioms for propositional logic is independent from each other. | It is an interesting question that whether some mathematical statement(s) can or cannot prove some other statement. To show a statement is provable, simply present a proof. But is there a way to show a statement is unprovable? For example, the Euclid's fifth postulate turned out to be unprovable from other postulates. Because one can find a geometry where all Euclid's postulates are ture (including the fifth) and another where all but the fifth are true. Since proof preserves truth one can say the fifth is unprovable. I am going to demonstrate a simple technique of showing the three axioms for propositional logic is independent from each other. | ||
| − | The axioms for proposition logic | + | The axioms for proposition logic are |
| − | 1. | + | 1. P -> (Q -> P) |
Revision as of 00:54, 5 September 2013
Independence of Axioms of Propositional Logic
Chenkai Wang
It is an interesting question that whether some mathematical statement(s) can or cannot prove some other statement. To show a statement is provable, simply present a proof. But is there a way to show a statement is unprovable? For example, the Euclid's fifth postulate turned out to be unprovable from other postulates. Because one can find a geometry where all Euclid's postulates are ture (including the fifth) and another where all but the fifth are true. Since proof preserves truth one can say the fifth is unprovable. I am going to demonstrate a simple technique of showing the three axioms for propositional logic is independent from each other.
The axioms for proposition logic are
1. P -> (Q -> P)
