Effectiveness in RPL, with applications to continuous logic

Abstract

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of Lukasiewicz logic) and continuous logic, and prove effective versions of some related theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; we show that provability degree of a formula with respect to a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an L p Banach lattice) is decidable.

Effective metric model theory

“…Consequently, the following proposition shows the equivalence of the approach of this article with what is in Didehvar and Ghasemloo (2009). Proposition 3.1.…”

“…This article is a further investigation of a project started in Didehvar and Ghasemloo (2009). In the aforementioned article, the authors prove an effective version of the completeness theorem for the rational Pavelka and the metric logics.…”

“…There are other frameworks for defining computability over the real numbers, such as domain theory which is used in Didehvar and Ghasemloo (2009). However, the TTE method is easier to implement for mathematical structures from analysis, and in particular, for the results proven in this paper.…”

“…Then, some basic notions of TTE are introduced. Afterwards, the concepts of computable and decidable metric structures are defined and some theorems of Didehvar and Ghasemloo (2009) which are needed throughout this article are quoted.…”

“…By giving some concrete effective constructions, it is shown that the separable atomless probability algebra and the rational Urysohn space are both decidable structures. It is known that both of these structures have separably categorical and computably axiomatizable theories and thus, are decidable structures (in light of Theorem 3.5 from Didehvar and Ghasemloo (2009)). However, no direct construction is given in Didehvar and Ghasemloo (2009) to show the computability of the structures.…”