most recent 30 from http://mathoverflow.net 2013-05-19T12:58:46Z http://mathoverflow.net/feeds/question/109415 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109415/could-this-peculiar-set-theory-be-of-any-interest-even-though-it-is-trivial Garabed Gulbenkian 2012-10-11T20:47:31Z 2012-10-11T21:01:13Z

<p>.......Let T denote the first order theory of dense linear order with no infimum and no supremum. The only atomic formulae in the language of T are "$x \lt y$" and "$x=y$" (where "x" and "y" are variables). This theory T is axiomatizable (it has about 6 or 7 axioms). T has no finite models and is categorical in the smallest infinite cardinal. Therefore, by a theorem of Vaught, T is consistent, complete and decidable. One would hardly call T a trivial theory, since many of its theorems are used to prove results in more general theories of ordered sets. .......The first order set theory T* is obtained from T by substituting the symbol for set membership for the symbol "$\lt$" wherever "$\lt$" occurs (in all the formulae of T). The axioms of T* are the transformed axioms of T after the substitution. The language of T* is the language of first order ZF. Then (although there may be a gross error here) I claim that T* shares with T the properties of being consistent, complete and decidable. Every sentence of first order ZF can be proved or disproved in T* (including all the sentences that are undecidable). If anybody wants to know what T* has to say about the Continuum Hypothesis, they can even work it out for themselves-because T* is decidable. Nevertheless, T* seems like the most trivial set theory that it is possible to have. What can one do with it? Not even the weakest sub-theory of arithmetic appears to be interpretable in it. What can one say about it? The set-theoretic universe of T* is infinite, yet not one of the sets that it contains is specifically definable within it. It violates the axiom of foundation in the worst possible way. It negates almost all axioms of comprehension that (in other set theories) are used to prove the existence of sets satisfying a variety of conditions. Last, but not least T* thumbs its nose at Godel"s incompleteness theorem. Can it be be that all trivial set theories are not created equal?</p>