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Group Structure on any set

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>So, what do you think ? Is it really possible to construct a model of ZF >with a non-empty set without any group structure? Hard to believe, but... Yes, this is possible; in fact, it happens in one of the first models one runs into when considering forcing constructions of models of ZF without choice. This is the model obtained by forcing in countably many Cohen reals and then taking a suitable symmetric submodel to get a model of ZF containing an infinite Dedekind-finite set of reals. The constructed set A in this model satisfies: (1) A is infinite; (2) there is no infinite sequence of distinct elements of A. The same method used to prove (2) can also be used to prove: (3) For any natural number n > 1, there is no partition of A into sets of size n. But a set A with these three properties cannot have a group structure. If it did, then by (1) there would be a non-identity element x. If x has infinite order, then the sequence of powers of x contradicts (2); if x has finite order, then

Group Structure on any set Groups Groups  Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable. Dismiss Learn more     Group Structure on any set 336 views Skip to first unread message  Faizal Ahmed unread, Jan 30, 2003, 6:29:10 AM 1/30/03   Delete You do not have permission to delete messages in this group Copy link Report message Show original message Either email addresses are anonymous for this group or you need the view member email addresses permission to view the original m

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