CEU Electronic Theses and Dissertations, 2013
| Author | Gyenis, Zalán |
|---|---|
| Title | Finite categoricity and non-atomicity of free algebras |
| Summary | This thesis consists of two parts. In the first part we are taking the first steps towards studying so called finitely categorical structures. By a celebrated theorem of Morley, a structure $\cA$ is $\aleph_1$-categorical if and only if it is $\kappa$-categorical for all uncountable $\kappa$. Our main goal is to examine finitary analogues of Morley's theorem. A model $\cA$ is defined to be finitely categorical (or $<\!\ omega$-categori cal) if for a large enough finite set $\Delta$ of formulae $\cA$ can have at most one $n$-element $\Delta$-elementary substructure for each natural number $n$. We are going to investigate some conditions on $\aleph_1$-categorical structures which imply finite categoricity. Proving finite categoricity for certain $\aleph_1$-categorical structures can be considered as an extension of Morley's theorem ``all the way down'' ;.\\ &#x 09;The second part of the present work deals with G\"odel's incompleteness property of logics. We show that the three-variable reduct of first order logic without equality but with substitutions or permutations has G\"odel's incompleteness property. An algebraic consequence of this is that the one-generated free three dimensional substitutional and polyadic algebras $\FrSCA$ and $\FrPA$ are not atomic. This provides a partial solution to a longstanding open problem of N\'emeti and Maddux going back to Alfred Tarski via the book \cite{TG}. |
| Supervisor | Sági, Gábor |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2013/gyenis_zalan.pdf |
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