In logic (and in mathematical logic) I wondered what does it mean to say that a given theory is weak or strong?
To be specific, I am just interested in arithmetics. Thanks!
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One theory is stronger than another if it allows more conclusions (or shorter proofs for the same conclusions) than another theory. For example, this is the result of adding more axioms to the weaker theory.
One theorem is stronger than another if it has a stronger conclusion from the same premise, or the same conclusion from a weaker premise.
ASCII Advocate
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Some authors call a theory "stronger" than another theory if it is capable of proving more theorems. For example, you could develop the theory of the natural numbers only allowing yourself axioms for what addition is. You could also develop a theory of the natural numbers with both addition and multiplication and how the interact. The second theory would be stronger than the first.
J126
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so, w.r logical systems, eg, SOL is stronger than FOL, right? – Bill Aug 09 '15 at 20:45
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@Bill You are correct. – J126 Aug 09 '15 at 22:56