{"id":12781,"date":"2009-03-01T13:02:23","date_gmt":"2009-03-01T13:02:23","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2009\/03\/01\/a-cantorian-argument-why-possible-worlds-cannot-be-maximally-consistent-sets-of-propositions-2\/"},"modified":"2009-03-01T13:02:23","modified_gmt":"2009-03-01T13:02:23","slug":"a-cantorian-argument-why-possible-worlds-cannot-be-maximally-consistent-sets-of-propositions-2","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2009\/03\/01\/a-cantorian-argument-why-possible-worlds-cannot-be-maximally-consistent-sets-of-propositions-2\/","title":{"rendered":"A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions"},"content":{"rendered":"

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 now.<\/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","protected":false},"excerpt":{"rendered":"

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 of the possible worlds,  is the maximally consistent set of true propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, … 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":[235,481,228],"tags":[],"class_list":["post-12781","post","type-post","status-publish","format-standard","hentry","category-modal-matters","category-set-theory","category-truth"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/12781","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=12781"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/12781\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=12781"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=12781"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=12781"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}