This semester I'm directing an independent study on modal logic with a couple of students with strong math background. Yesterday some questions about sets and possible worlds came up, so I wrote up some notes for my students on the subject. This blog post is adapted from those notes.
With that in mind, we ask three questions: (1) could possible worlds be sets of sentences? (2) Could possible worlds be sets of propositions? (3) Could there be a set of all possible worlds? In all three cases, there are reasons to suspect that the answer is 'no'. This doesn't create any problem for using the possible worlds semantics as a heuristic for modal reasoning, or as a formal model to show the consistency of our modal logic, but it does create problems for trying to use possible worlds to give an analysis of necessity.
First, by Gödel Numbering. According to the Fundamental Theorem of Arithmetic, every natural number has a unique prime factorization. Assign a natural number to each symbol in our language. (We can do this, since we assumed there were at most countably many symbols.) Let the sequence of primes stand for the positions in the sentence. Then raise each prime to the power of the number for the symbol that belongs there. So, for instance, 5 is the 3rd prime. If we assign the symbol '&' to the number 2, and '&' is the 3rd symbol in our sentence, then we'll have 5^2 in the prime factorization of the number for the sentence. This allows us to assign a unique natural number to every sentence in our language, which shows that there are (at most) as many sentences as the natural numbers.
Second, by effective enumerability. A set is effectively enumerable iff there is an algorithm that will iterate over all the members of the set without missing any. If the set is infinite then the algorithm will never terminate, but you'll get to any member of the set if you run the algorithm long enough. Any effectively enumerable set is countable, since every member of the set must be the nth to be produced by the algorithm for some natural number n. Now, suppose for a moment that the set of basic symbols is finite. Then we can have our algorithm first produce all the sentences of length 1, then all those of length 2, and so forth. For each length, there are only finitely many sentences, so we won't get stuck and will eventually get through all of our sentences.
If we have a countable infinity of basic symbols the enumeration is a little more complicated, but we can still do it. Since we have countably many symbols, we can put them in some arbitrary order. Then we can get through all of them using the same iteration procedure used to prove the countability of the rationals. (See this picture.) So we first generate all the sentences of length 1 that use only the first symbol, then the sentences of length 2 using only the first symbol, then the sentences of length 1 using the first 2 symbols, then the sentences of length 1 using the first 3 symbols, and so on through the diagram. (The numerator is the length of the sentences, the denominator is the number of symbols.) In this way, we'll generate all possible sentences.
Ok, so there's a set - and a countable set at that! - of all the sentences in a given language. We might therefore try to define possible worlds as maximal consistent sets of sentences in some language. ('Consistent' meaning there's no contradiction; 'maximal' meaning if we added another sentence we'd get a contradiction.) But there's a problem: we don't seem to have enough sentences for all the ways the world could be. For instance, it seems possible that there could be point particles with perfectly precise locations in Euclidean space (even though our world is apparently not like this). Then, for every real number x, there should be a possible world where all that exists are two particles exactly x meters apart. But we don't have enough sentences to distinguish all of these worlds from one another. (There are theories about space that would allow you to deny that these worlds are possible, but there are also other such examples that could be constructed. Of course, you could try to reject all of them.)
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