{"id":11473,"date":"2010-07-10T13:31:33","date_gmt":"2010-07-10T13:31:33","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2010\/07\/10\/a-cantorian-argument-why-possible-worlds-cannot-be-maximally-consistent-sets-of-propositions\/"},"modified":"2010-07-10T13:31:33","modified_gmt":"2010-07-10T13:31:33","slug":"a-cantorian-argument-why-possible-worlds-cannot-be-maximally-consistent-sets-of-propositions","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2010\/07\/10\/a-cantorian-argument-why-possible-worlds-cannot-be-maximally-consistent-sets-of-propositions\/","title":{"rendered":"A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions"},"content":{"rendered":"
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A commenter in the 'Nothing' thread<\/a> spoke of possible worlds as sets.  What follows is a reposting from 1 March 2009 which opposes that notion.<\/font><\/p>\n

…………….<\/font><\/p>\n

\"CANTOR_OCT20_G_290w_q30\"<\/a> In a <\/font>recent comment<\/font><\/a>, Peter Lupu bids us construe possible worlds as maximally consistent sets<\/em> of propositions.  If this is right, then the actual world, which is of course one of the possible worlds,  is the maximally consistent set of true<\/em> propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.<\/font><\/p>\n

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1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. The power set of a set S is the set whose elements (members) are all of S's subsets. Recall the difference between a member and a subset. The set {Socrates, Plato} has exactly two elements, neither of which is a set. Since neither is a set, neither is a subset of this or any set. {Socrates, Plato} has four subsets: the set itself, the null set, {Socrates}, {Plato}. Note that none of the four sets just listed are elements of {Socrates, Plato}. The power set of {Socrates, Plato}, then, is {{Socrates, Plato}, { }, {Socrates}, {Plato}}.<\/font><\/p>\n

In general, if a set S has n members, then P(S) has 2n<\/sup> members. Hence the name power set. Cantor's Theorem that the power set of a set S is always strictly larger that S is easily proven. But the proof needn't concern us.  It is available in any standard book on set theory.<\/font><\/p>\n

2. Suppose there is a set T of all truths, {t1<\/sub>, . . . , ti<\/sub>, ti + 1<\/sub>, . . .}. Consider the power set P(T) of T. The truth t1<\/sub> in T will be a member of some of T's subsets but not of others. Thus, t1<\/sub> is an element of {t1<\/sub>, t2<\/sub>}, but is not an element of { }. In general, for each subset s in the power set P(T) there will be a truth of the form t1<\/sub> belongs to s<\/em> or t1<\/sub> does not belong to s.<\/em> But according to Cantor's Theorem, the power set of T is strictly larger than T. So there will be more of those truths than there are truths in T. It follows that T cannot be the set of all<\/em> truths.<\/font><\/p>\n

3. Given that there cannot be a set of all truths, the actual world cannot be the set of all truths.  This implies that possible worlds cannot be maximally consistent sets of propositions.   I learned the Cantorian argument that there  is no set of all truths from <\/font>Patrick Grim<\/font><\/a>. I don't know whether he applies it to the question whether worlds are sets.<\/font><\/p>\n

4. As far as I can see, the fact that possible worlds cannot be maximally consistent sets<\/em> does not prevent them from from being maximally consistent conjunctive propositions<\/em><\/font>.<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

A commenter in the 'Nothing' thread spoke of possible worlds as sets.  What follows is a reposting from 1 March 2009 which opposes that notion. ……………. In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions.  If this is right, then the actual world, which is of course one … Continue reading “A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[688,235,481],"tags":[],"class_list":["post-11473","post","type-post","status-publish","format-standard","hentry","category-cantor","category-modal-matters","category-set-theory"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/11473","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/comments?post=11473"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/11473\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=11473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=11473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=11473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}