Elementary equivalence of ordinals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:38:36Z http://mathoverflow.net/feeds/question/35971 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35971/elementary-equivalence-of-ordinals Elementary equivalence of ordinals Ricky Demer 2010-08-18T14:20:11Z 2010-08-18T17:25:14Z <p>What is the smallest ordinal alpha which is elementarily equivalent to some smaller ordinal beta with the signature (&lt;)?</p> <p>What is the corresponding ordinal beta?</p> <p>What if we instead require that beta be an elementary substructure of alpha?</p> http://mathoverflow.net/questions/35971/elementary-equivalence-of-ordinals/35982#35982 Answer by Andreas Blass for Elementary equivalence of ordinals Andreas Blass 2010-08-18T16:23:13Z 2010-08-18T16:23:13Z <p>The first-order theory of well-orderings was studied in great detail in a paper of Doner, Mostowski, and Tarski, "The elementary theory of well-ordering -- a metamathematical study" [Logic Colloquium '77, edited by A. Macintyre, L. Pacholski, and J. Paris, North-Holland (1978) pp. 1-54]. In particular, their Corollary 44 characterizes (unless their notation is very non-standard --- I haven't checked carefully) when two ordinals are elementarily equivalent. Modulo an apparent typo in the definition just before the corollary (one of the strict inequalities should be non-strict), it seems that the first pair of distinct but elementarily equivalent ordinals is <code>$\omega^\omega$</code> and <code>$\omega^\omega\cdot2$</code>. A thorough reading of the paper (which I don't have time for right now) should also reveal the answer to your second question, about elementary submodels (probably the same pair of ordinals). </p>