Impredicative Theory Research Articles

Epsilon substitution for $$\textit{ID}_1$$ ID 1 via cut-elimination

The \(\epsilon \)-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory \(\textit{ID}_1\) using a variant of the cut-elimination formalism introduced by Mints.

Goodstein sequences for prominent ordinals up to the ordinal of [formula omitted

We introduce strong Goodstein principles which are true but unprovable in strong impredicative theories like IDn.

Sublocales in formal topology

AbstractThe paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the collection of inductively generated sublocales has coframe structure.Overt sublocales and weakly closed sublocales are described, and related via a new notion of “rest closed” sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with “lower powerpoints”, arising from the impredicative theory of the lower powerlocale.Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with “upper powerpoints”, arising from the impredicative theory of the upper powerlocale.

How to develop Proof‐Theoretic Ordinal Functions on the basis of admissible ordinals

AbstractIn ordinal analysis of impredicative theories so‐called collapsing functions are of central importance. Unfortunately, the definition procedure of these functions makes essential use of uncountable cardinals whereas the notation system that they call into being corresponds to a recursive ordinal. It has long been claimed that, instead, one should manage to develop such functions directly on the basis of admissible ordinals. This paper is meant to show how this can be done. Interpreting the collapsing functions as operating directly on admissible sets also renders a new and perspicuous approach to well‐ordering proofs possible. MSC: 03F15, 03F35.

A theory for program and data type specification

The theory presented here, IOCC (Impredicative theory of Operations, Control and Classes) is an essentially first-order, two-layered theory. The lower layer is a theory of program equivalence and definedness. Program primitives that can be treated within this theory include functional application and abstraction, conditional, numbers, pairing, and continuation capture and resumption. The upper layer is a theory of class membership with a general comprehension principle for defining classes. Many standard class constructions and data types can be represented naturally including algebraic abstract data types, list-, record-, and function-type constructions. In addition classes can be constructed satisfying systems of equations such as[ S = A× G; G = B → S] which are useful in treating programming concepts such as streams and object behaviors. Coroutines can also be classified within this theory. Examples are given illustrating the derivation of programs using escape mechanisms, and methods for reasoning about streams and coroutines. Methods are given for constructing models of IOCC from semantic models of the underlying programming language.

Two Impredicative Theories of Properties and Sets

Two Impredicative Theories of Properties and Sets

Forcing for the impredicative theory of classes

The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .