Are There Set Theoretic Possible Worlds? Author(s): Selmer Bringsjord Source: Analysis, Vol. 45, No. 1, (Jan., 1985), pp. 64 Published by: Blackwell Publishing on behalf of The Analysis Committee Stable URL: http://www.jstor.org/stable/3327409 Accessed: 17/04/2008 19:24 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
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ARE THERE SET THEORETIC POSSIBLEWORLDS? By SELMERBRINGSJORD
HERE are of course, many conceptions of (possible) worlds
currently in fashion. In this note I show that one of these conceptions (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, Quantification, 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 p C w or -p E w; and (ii) the members of w are 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 2k; and 2k> k. But for each element ei C 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 (D1) 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 2k. But we started with the assumption that the cardinal number of w is k.1
? SELMER BRINGSJORD 1985
Brown University, Providence, Rhode Island 02912, U.S.A.
'I am indebted to Roderick Chisholm, Michael Zimmerman, Atli Hardarson, and an anonymous referee.
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