Models and Ultraproducts : An Introduction

TL;DR: The paper establishes the general structure of the inconsistent models of arithmetic of [7], and it is shown that such models are constituted by a sequence of nuclei, which can have the order type of any ordinal, of therationals, or of any other order type that can be embedded in the rationals in a certain way.

Abstract: In [2] N. C. A. da Costa surveys some interesting results about inconsistent formal systems. A formal system is said to be inconsistent if there is a formula φ such that both φ and~<ρ are theorems. The approach in [2] towards the study of inconsistent systems is basically syntactical. In this paper we investigate inconsistent theories from a model-theoretical point of view. However we do not analyze semanticaily the calculi presented in [2] as suggested on Page 508. Instead we define a notion of structure which allows for the possibility of built-in inconsistencies. These structures may then be models of inconsistent theories. We classify theories in 3 different ways. Intuitively, the higher a theory is in a classification, the more inconsistent it is. This way we obtain measures of inconsistency for theories.

TL;DR: In contrast with other semantic theories, the authors proposes that the interpretation of the comparative morpheme remains the same whether it appears in sentences that compare individuals directly or indirectly, and suggests that all comparisons involve a scale of universal degrees that are isomorphic to the rational (fractional) numbers between 0 and 1.

TL;DR: It is consistent that there does not exist a universal space (either by embeddings or by mappings onto) in , and it is proved that there exists a space X ∈ which is universal in the sense of embedDings.