lo.logic - Are all mathematical theorems necessarily true?

Charles wrote:

Robert Hanna (Kant's Theory of Judgment, SEP 2009, sect. 2.2.2) interprets Kant as saying that "logically possible worlds are nothing but maximal logically consistent sets of concepts". If we apply this by saying that the logically possible worlds that can contain the concepts as expressed by an axiomatisation are the models of that theory, then it follows that the theorems of an axiomatisation are necessarily true.

The logical quantifiers interact in odd ways with this idea, and make "maximality" kind of a problematic notion. This is a familiar phenomenon for mathematicians and logicians, of course, but may be worth pointing out explicitly. So, if we have some set of atomic concepts or assertions about the world, then a model of a possible world is a Boolean algebra on this set. Then the basic logical connectives (conjunction, disjunction, negation) can be modelled by intersection, union and complement.

So far, so good. Now, if the set of atomic assertions is finite, then the Boolean algebra is finite, and so it is also a complete lattice, and so quantified statements also have interpretations in the model. But if the set of atomic assertions is not finite, then a Boolean algebra on this set doesn't have to be complete, and so quantified statements might not have interpretations! If we do demand that the Boolean algebra is complete, then what we consider to be "logically possible" depends upon what kinds of logical connectives we wish to consider. (For example, in probability theory the difference between the Kolmogorov and Bayesian axioms of probability is that the Bayesians don't demand countable additivity, which means that an existentially-quantified propositions might not have a Bayesian interpretation.)