Simple Models of Randomization and Preservation Theorems
Abstract
The main purpose of this paper is to present new and more uniform model-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability is a (complete) first-order continuous theory with $NIP$/stability. The proof method for both theorems is based on the significant use of a particular type of models of $T^{R}$, namely simple models, and certain indiscernible arrays. Using simple models of $T^R$ gives the advantage of re-proving these theorems in a simpler and quantitative manner. We finally turn our attention to $NSOP$ in randomization. We show that based on the definition of $NSOP$ given [11], $T^R$ is stable if and only if it is $NIP$ and $NSOP$.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.15014
- arXiv:
- arXiv:2408.15014
- Bibcode:
- 2024arXiv240815014K
- Keywords:
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- Mathematics - Logic;
- Mathematics - Combinatorics
- E-Print:
- 21 pages. Comments welcome. k.khanaki @ gmail.com