Generalized geometric theories and set-generated classes

We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is thatthe class of models of a generalized geometric theory is set-generated. Here, a class$\mathcal{X}$of subsets of a set is set-generated if there exists a subsetGof$\mathcal{X}$such that for each α ∈$\mathcal{X}$, and finitely enumerable subset τ of α there exists a subset β ∈Gsuch that τ ⊆ β ⊆ α. We show the main result in the constructive Zermelo–Fraenkel set theory (CZF) with an additional axiom, called the set generation axiom which is derivable inCZF, both from the relativized dependent choice scheme and from a regular extension axiom. We give some applications of the main result to algebra, topology and formal topology.

The disjunction and related properties for constructive Zermelo-Fraenkel set theory

This paper proves that the disjunction property, the numerical existence property. Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.

Frege proof system and TNC°

A Frege proof systemFis any standard system of prepositional calculus, e.g., a Hilbert style system based on finitely many axiom schemes and inference rules. An Extended Frege systemEFis obtained fromFas follows. AnEF-sequence is a sequence of formulas ψ1, …, ψκsuch that eachψiis either an axiom ofF, inferred from previous ψuand ψv(= ψu→ ψi) by modus ponens or of the formq↔ φ, whereqis an atom occurring neither in φ nor in any of ψ1,…,ψi−1. Suchq↔ φ, is called an extension axiom andqa new extension atom. AnEF-proof is anyEF-sequence whose last formula does not contain any extension atom. In [12], S. A. Cook and R. Reckhow proved that the pigeonhole principlePHPhas a simple polynomial sizeEF-proof and conjectured thatPHPdoes not admit polynomial sizeF-proof. In [5], S. R. Buss refuted this conjecture by furnishing polynomial sizeF-proof forPHP. Since then the important separation problem of polynomial sizeFand polynomial sizeEFhas not shown any progress.In [11], S. A. Cook introduced the systemPV, a free variable equational logic whose provable functional equalities are ‘polynomial time verifiable’ and showed that the metamathematics ofFandEFcan be developed inPVand the soundness ofEFproved inPV. In [3], S. R. Buss introduced the first order systemand showed thatis essentially a conservative extension ofPV. There he also introduced a second order system(BD).