What can be said about uncountable propositional language? And how much can it be said?
Well, when I approach the primer of propositional logic, it usually begins with describing the syntax of the language, i.e.,
1. specifying the set PROP of propositional variables and the logical connectives to be used in the study;
2. specifying the rules about how to construct the propositional formulas.
Accompanying such description is the assumption (made by the instructor or the author of the book) that the set PROP in the study will be denumerable (so the resulted propositional language is said to be denumerable). Then comes a sentence that irritates me: the study can be easily generalized to the uncountable language (the language with an uncountable set PROP).
I've been wondered about that word, easily. I know that the big metatheorems (Completeness, Compactness) and other minor metatheorems still hold in the uncountable case. A slight distinction between the two versions of languages perhaps lies in a proof of Lindenbaum's theorem ("a consistent set of formulas can be extended to a complete theory"), where in the uncountable language, Zorn's lemma must be invoked, wheareas nothing subtle happens in the denumerable case. But is it all that? Does the word easily mean there's no more subtle issue, those worth mentioning issue, lurking behind the careful analysis of the two versions of languages? The following are some examples of what I've been curious about:
1. The cardinality of the set FORM of propositional formulas in the uncountable language.
2. The number of truth assignments (any function f:PROP--->{0,1}) in the uncountable language.
3. The notion of independent axiomatizability in the uncountable language. A well-known theorem is that any set (finite or denumerable) of propositional formulas is independently axiomatizable.
(A set S of formulas is independently axiomatizable if S=Cn(T) for some independent set T of formulas, where Cn(T) denotes the set of formulas semantically implied by T.)
Is anyone has an idea or can explain me about this?
PS. What makes me strongly believe that there must be some "non-easy" distinctions between different languages built up from different choices of PROP comes from my study of the Lindenbaum-Tarski algebra of propositional language. On the study, I paid my attention to the case where the propositional languages are finite and denumerable, respectively. What I obtained is that in the denumerable language (which every one and every textbook are so delighted to work with), the Lindenbaum-Tarski algebra is an atomless, denumerable, free Boolean algebra. But believe it or not, when I restricted the study to language built up from the finite set PROP, the algebra turns out to be an atomic, finite, free Boolean algebra! (I'm sorry for not explaining the definitions of the Lindenbaum-Tarski algebra, atomless Boolean algebras, atomic Boolean algebras, free Boolean algebras. These can be "easily" found in Google.) I'm still wondering what the Lindenbaum-Tarski algebra will turn to be when I assume that PROP is uncountable!