{"product_id":"forcing-isomorphisms-between-dense-sets-of-reals-von-michael-h-vartanian","title":"Forcing Isomorphisms Between Dense Sets of Reals","description":"\u003cp\u003eIn founding set theory, Cantor showed that the cardinality of the set Q of rational numbers is countably infinite;  that Q may be extended by completion to obtain the set R of  real numbers (we say that Q is countably dense in R);  that any other countably dense subset of R is isomorphic to Q; and that R itself is uncountably infinite. The question then naturally arises whether all uncountably dense subsets of R of the same cardinality must also be isomorphic. Decades later, a negative answer was given when a model of set theory was constructed in which many uncountably dense subsets of R fail to be isomorphic. On the other hand, Baumgartner has shown by the method of forcing that another model exists in which all dense subsets of R of the least uncountable cardinality are isomorphic. Presented here is a detailed yet expository account of Baumgartner's famous result with a brief discussion of its relevance to forcing axioms in contemporary set theory.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783846528891\"\u003e\u003ch3\u003eA Classic Result of Modern Set Theory\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783846528891","offer_id":39494498680925,"sku":"9783846528891","price":49.0,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/5decfd9e-71f6-45c4-8234-20a64b509a0d.jpg?v=1758346125","url":"https:\/\/shop.autorenwelt.de\/en\/products\/forcing-isomorphisms-between-dense-sets-of-reals-von-michael-h-vartanian","provider":"Autorenwelt Shop","version":"1.0","type":"link"}