ARE THERE SET THEORETIC POSSIBLE WORLDS?
By
SELMER BRINGSJORD
HERE are of course, many conceptions of (possible) worlds
currently in fashion. In this note I show that one of these conT
ceptions (viz., one which identifies worlds with sets of states of
affairs or propositions) is formally incoherent. (My proof is somewhat similar to a paradox Martin Davies presents in his Meaning,
Quanttfication, Necessity (Routledge and Kegan Paul, 1981),
Appendix 9. However, unlike Davies, I make no use of the rather
slippery notion of thinking a proposition. And, though Davies
perhaps shows that there is no such thing as the set of all worlds,
I think I show that there is simply no such thing as a world, in the
set theoretic sense.)
The set-theoretic account I attack herein is the following (it
encapsulates a synthesis of those set-theoretic construals of possible
worlds championed by Robert Adams and Alvin Plantinga):
(Dl) w is a world ==df. w is a set of states of affairs such that
(i) for every state of affairs p: either pEw or ~p E w;
and (ii) the members of ware compossible.
The proof is as follows. Take any world w; and assume that the
cardinal number of w is k. Consider, next, the power set of w,
which we will denote by 'P(w)'. The cardinality of P(w), by
Cantor's theorem, will be 2 k ; and 2 k > k. But for each element
ei E P(w) there will be a corresponding state of affairs (or, if you
like, a proposition) having the form ei's being a set. (Any old state
of affairs will do here: ei's being an element of P(W) , etc.) And it
follows from (D 1) that for each of these states of affairs, either it or
its negation will be an element of w. Hence the cardinal number of
w is at least 2 k • But we started with the assumption that the cardinal
number of w is k. 1
Brown University,
Providence, Rhode Island 02912,
U.S.A.
©
SELMER BRINGSJORD
1985
1 I am indebted to Roderick Chisholm, Michael Zimmerman, Atli Hardarson, and an
anonymous referee.
64