Smullyan's truth and provability

TL;DR

This paper revisits Smullyan's 2013 work on truth and provability, introducing models, clarifying relationships between theorems, and constructing models linked to arithmetic properties.

Contribution

It introduces Smullyan models, clarifies theorem relationships, and constructs models connecting truth, provability, and arithmetic.

Findings

Defined Smullyan models precisely.

Clarified theorem implications and non-implications.

Constructed models linking truth and provability in arithmetic.

Abstract

We revisit Smullyan's paper ``Truth and Provability'' (2013) for three purposes. First, we introduce the notion of Smullyan models to give a precise definition for Smullyan's framework discussed in that paper. Second, we clarify the relationship between three theorems proved by Smullyan and other newly introduced properties for Smullyan models in terms of both implications and non-implications. Third, we construct two Smullyan models based on arithmetical ideas and show the correspondence between the properties of these Smullyan models and those concerning truth and provability in arithmetic.