arXiv subjects
Publications and source records attributed to Seyed-Mohammad Bagheri.
Topological semantics for affine modal propositional logic is introduced. The interior operator on subsets is replaced with the lower semi-continuous envelope operator on functions. Completeness and affine compactness theorems are proved for this logic.
Affine continuous logic is extended to affine integration logic. Affine compactness theorem is proved by both the ultramean construction and Henkin's method. Also, a proof system and a completeness theorem are given. An appropriate variant of the Keisler-Shelah isomorphism theorem holds in this setting. This helps us to characterize non-forking extensions in affine stable theories by means of the notion of elementary embedding in the expanded logic.
By affine arithmetic is meant the set of affine consequences of Peano arithmetic. This is a continuous theory which is studied in the framework of affine logic, a sublogic of continuous logic. Affine arithmetic is undecidable. Also, its models are generally lattice ordered and carry a nontrivial metric. Classical models are then characterized as those which are linearly ordered. In this paper, the affine variants of several classical results in Peano arithmetic are proved. In particular, an affine form of Gaifman's splitting theorem is proved.
Quantifier-elimination or model-completeness of the affine part of some classical first order theories are proved.
By Lindström's theorems, the expressive power of first order logic (and similarly continuous logic) is not strengthened without losing some interesting property. Weakening it, is however less harmless and has been payed attention by some authors. Affine continuous logic is the fragment of continuous logic obtained by avoiding the connectives $\wedge,\vee$. This reduction leads to the affinization of most basic tools and technics of continuous logic such as the ultraproduct construction, compactness theorem, type, saturation etc. The affine variant of the ultraproduct construction is the ultramean construction where ultrafilters are replaced with maximal finitely additive probability measures. A consequence of this relaxation is that compact structures with at least two elements have now proper elementary extensions. In particular, they have non-categorical theories in the new setting. Thus, a model theoretic framework for study of such structures is provided. A more remarkable aspect of this logic is that the type spaces are compact convex sets. The extreme types then play a crucial role in the study of affine theories. In this text, we present the foundations of affine continuous model theory.
I study definable sets in affine continuous logic. Let $T$ be an affine theory. After giving some general results, it is proved that if $T$ has a first order model, its extremal theory is a complete first order theory and first order definable sets are affinely definable. In this case, the type spaces of $T$ are Bauer simplices and they coincide with the sets of Keisler measures of the extremal theory. In contrast, if $T$ has a compact model, definable sets are exactly the end-sets of definable predicates. As an example, it is proved in the theory of probability algebras that one dimensional definable sets are exactly the intervals $[a,b]$.
In the affine fragment of continuous logic, type spaces are compact convex sets. I study some model theoretic properties of extreme types. It is proved that every complete theory $T$ has an extremal model, i.e. a model which realizes only extreme types. Extremal models form an elementary class in the full continuous logic sense if and only if the set of extreme $n$-types is closed in $S_n(T)$ for each $n$. Also, some applications are given in the special cases where the theory has a compact or first order model.
The ultraproduct construction is generalized to $p$-ultramean constructions ($1\leqslant p<\infty$) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments $\mathscr L^p$ of continuous logic. A powermean variant of Keisler-Shelah isomorphism theorem is proved for $\mathscr L^p$. It is then proved that $\mathscr L^p$-sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.
We combine continuous and integral logics and found a logical framework for metric measure spaces equipped with a family of continuous relations and operations. We prove the ultraproduct theorem and deduce compactness and other usual results. We also give applications of the compactness theorem in metric measure theory.
This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the Eberlein-Smulian compactness theorem by Ramsey's theorem and point out the correspondence between this theorem and a result in Shelah's classification theory.