2 results
KEISLER’S ORDER IS NOT LINEAR, ASSUMING A SUPERCOMPACT
-
- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 2 / June 2018
- Published online by Cambridge University Press:
- 01 August 2018, pp. 634-641
- Print publication:
- June 2018
-
- Article
- Export citation
-
We show that if there is a supercompact cardinal, then Keisler’s order is not linear. More specifically, let Tn,k be the theory of the generic n-clique free k-ary graph for any n > k ≥ 3, and let TCas be the simple nonlow theory described by Casanovas in [2]. Then we show that TCas
$$Tn,k always, and if there is a supercompact cardinal then Tn,k
$$TCas.
THE NUMBER OF ATOMIC MODELS OF UNCOUNTABLE THEORIES
-
- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 1 / March 2018
- Published online by Cambridge University Press:
- 08 February 2018, pp. 84-102
- Print publication:
- March 2018
-
- Article
- Export citation
-
We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with
$ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that there is a complete theory in a language of size 
${\aleph _1}$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with 
$ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that for every complete theory T in a language of size 
${\aleph _1}$, if T has uncountable atomic models but no constructible models, then T has 
${2^{{\aleph _1}}}$ atomic models of size 
${\aleph _1}$.
