I just read an extract of David Lewis's Counterfactuals and he claims there in a footnote on page 90 that there are at least beth_2 possible worlds. He also claims in the very same footnote that "[i]t can easily be shown that this is the number of ersatz worlds in Quine’s original construction." (Lewis 1973, p. 90) Would anyone know if there is literature on this statement (maybe even a proof of this proposition) and more generally on the cardinality of the logical space? Thank you in advance for your help!
It seems strange to me. Firstly, we would need to specify what type of language is D. Lewis considering. In addition, it is necessary to specify what is a word within such language.
To simplify, a language is based on combinations of letters of a certain alphabet (written languages have finite alphabets and spoken languages have a finite number of distinct phonemes). With a finite alphabet, a word is a finite sequence of symbols of the alphabet, the possible descriptions in such a language are infinite. Specifically, the inifnite set of finite sequences of words in such a language has cardinality ℵ0 = ℶ0. The the beth numbers are defined recursively as:
ℶn+1 = 2ℶn
So, this number is equivalent to the possible infinite collections of sentences in the specified language (but these sets are, themselves not expressible by means of the language). This situation appears in formal languages. For example, using the current mathematical notation we can specify an infinite collection of entities, with cardinality ℶ0 (equals ℵ0). This means for example, that the vast majority of real numbers cannot be "referred to" using the conventional notation, because there are ℶ1 real numbers, but only ℶ0 possible expressions to denote the entities and ℶ0 < ℶ1.
On the other hand, ℶ2 = 2ℶ1 is another amazingly large infinite number/cardinal, which of course, includes many more entities than can be named with ordinary formal language.