A logic-enriched type theory is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named and , which we claim correspond closely to the classical predicative systems of second order arithmetic and . We justify this claim by translating each second order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches (...) to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory , which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system has also been claimed to correspond to Weyl’s foundation. By casting and as LTTs, we are able to compare them with . It is a consequence of the work in this paper that is strictly stronger than .The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work. (shrink)

We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate ($\models^*$) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension $\Theta$ of ZF there is a finitely axiomatizable extension $\Theta'$ of GB that is a conservative extension of $\Theta$. We also (...) prove a conservative extension result that justifies the use of $\models^*$ to characterize ground models for forcing constructions. (shrink)

We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility.

In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of (...) the same formula. We argue that the first makes use of a too strong metatheory. Although meaningful, this symmetrical reduction does not equate the ontological content of the theories. The strong metatheory levels the two theories. Moreover, we will modernize Shoenfields proof, emphasizing its relation to Herbrands theorem and that it can only be seen as a partial type of reduction. In contrast with symmetrical reductions, we believe that asymmetrical relations are powerful tools for comparing ontological content. In virtue of this, we prove that there is no interpretation of NBG in ZF, while NBG trivially interprets ZF. This challenges the standard view that the two systems have the same ontological content. (shrink)

Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions (...) can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Gödel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems. (shrink)

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper (...) classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems. (shrink)