A thought on logic
Apr. 8th, 2008 03:58 pmThis is something that had been bothering me.
When I first read about model theory, saw the definition of a model, read that if a set of axioms has a model then it's consistent... "But... doesn't that assume the axioms of set theory are themselves consistent?"
And then you have things like the compactness and completeness theorems, which require axiom of choice - but this is a fact of set theory! Logic can't be based on it; and what if we use a different set theory? Logic doesn't require set theory, so how can this logical "theorem" follow? (And meanwhile we're always working with a set of symbols for our alphabet...)
This had been bothering me for quite a while. Recently, though, I realized how to resolve the problem: There are two different things here called "logic". One is the formal logic that math is based on, where axioms are made into other statements by means of given rules. The other is the recreation of logic within math (i.e. within set theory), where we define a statement as such and such an object, etc, much as we model computability within math. This allows us to talk about axiomatizability and such, and prove the compactness theorem, and assume we have the set-theoretic universe, and talk about model theory in a way that makes sense. But this only applies to logic-within-math, not to logic-without-math. I haven't seen any distinction drawn between them before, but I don't see how model theory can make any sense without it. Is this a generally recognized distinction that just hadn't made its way to me, or are things really as muddled as they appear?
-Harry
When I first read about model theory, saw the definition of a model, read that if a set of axioms has a model then it's consistent... "But... doesn't that assume the axioms of set theory are themselves consistent?"
And then you have things like the compactness and completeness theorems, which require axiom of choice - but this is a fact of set theory! Logic can't be based on it; and what if we use a different set theory? Logic doesn't require set theory, so how can this logical "theorem" follow? (And meanwhile we're always working with a set of symbols for our alphabet...)
This had been bothering me for quite a while. Recently, though, I realized how to resolve the problem: There are two different things here called "logic". One is the formal logic that math is based on, where axioms are made into other statements by means of given rules. The other is the recreation of logic within math (i.e. within set theory), where we define a statement as such and such an object, etc, much as we model computability within math. This allows us to talk about axiomatizability and such, and prove the compactness theorem, and assume we have the set-theoretic universe, and talk about model theory in a way that makes sense. But this only applies to logic-within-math, not to logic-without-math. I haven't seen any distinction drawn between them before, but I don't see how model theory can make any sense without it. Is this a generally recognized distinction that just hadn't made its way to me, or are things really as muddled as they appear?
-Harry
