A constructive naive set theory and the $\omega$-rule

We prove that Constructive Naive Set Theory CONS, a naive set theory within Full Lambek predicate calculus with exchange and weakening rule FLew$¥forall$ does not prove the crispness, i.e. tertium non datur holds, for $\omega$ which is a set of natural numbers.

##The final version will be to appear in Notre Dame Journal of Formal Logic## pComp, a para-complete naive set theory in FLew∀ (intuition-istic logic minus the contraction rule) is consistent and we can develop its metamathematics using itself. The significance of pComp is that it allows circular definitions of very strong form, though it is proof theoretically weak. However, the details of such circularly defined sets are not well-known: we do not know whether they contain non-standard elements in particular. In this paper, as a testbed, we investigate the non-standardness of ω, the set of natural numbers which is also defined circularly, and we give negative answers to the problem of whether pComp is ω-consistent, using co-inductive objects essentially.

Review of Symbolic Logic

We give some negative results on the expressiveness of naïve set theory (NS) in LP, and in the four variants of minimally inconsistent LP defined in Crabbé : LPm, LP=, LP⊆ and LP⊇. We show that NS in LP cannot prove the existence of sets which behave like singleton sets, Cartesian pairs, or infinitely ascending linear orders. We show that NS is close to trivial in LPm and LP⊆, in the sense that its only minimally inconsistent model is a one-element model. We show that NS in LP= and LP⊇ has the same limitations we give for NS in LP. §1. Introduction. By naïve set theory (NS), we mean the set theory consisting of extensionality and comprehension:

arXiv (Cornell University), 2022

We survey the logical structure of constructive set theories and point towards directions for future research. Moreover, we analyse the consequences of being extensible for the logical structure of a given constructive set theory. We finally provide examples of a number of set theories that are extensible.

Annals of Pure and Applied Logic, 1984

Logical Methods in Computer Science, 2007

We propose a set theory strong enough to interpret powerful type theories underlying proof assistants such as LEGO and also possibly Coq, which at the same time enables program extraction from its constructive proofs. For this purpose, we axiomatize an impredicative constructive version of Zermelo-Fraenkel set theory IZF with Replacement and ω-many inaccessibles, which we call IZFRω. Our axiomatization utilizes set terms, an inductive definition of inaccessible sets and the mutually recursive nature of equality and membership relations. It allows us to define a weakly-normalizing typed lambda calculus corresponding to proofs in IZFRω according to the Curry-Howard isomorphism principle. We use realizability to prove the normalization theorem, which provides a basis for program extraction capability.

2021

In this article Russell's paradox and Cantor's paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.

Dialectica, 2008

Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit 'interpreting' instances that make the implication valid. For proofs in constructive set theory CZF-, it may not always be possible to find just one such instance, but it must suffice to explicitly name a set consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧-interpretation theorem for constructive set theory in all finite types CZF wshows. By changing to a hybrid interpretation ∧q, we show closure of CZF wunder rules that-in stronger forms-have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZF wand its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation.

Hay's rule gives rise to ω-inconsistency in conjunction with unrestricted abstraction and a contraction free logic weaker than Lukasiewicz' infinite-valued logic.

2001

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form V κ where κ is a strongly inaccessible cardinal and V κ denotes the κ -th level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depends on the context in which they are embedded. The context here will be the theory CZF − of constructive Zermelo Fraenkel set theory but without ∈ -Induction (foundation). Let INAC be the statement that for every set there is an inaccessible set containing it. CZF − + INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF − + INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF − + INAC is a small ordinal known as the Feferman -Schütte ordinal Γ 0 .