Constructing the Constructible Universe Constructively

We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the Axiom of Exponentiation holds in L.Comment: 26 pages. Revised following referee's recommendation

Power Kripke–Platek set theory and the axiom of choice

While power Kripke–Platek set theory, KP(P)⁠, shares many properties with ordinary Kripke–Platek set theory, KP⁠, in several ways it behaves quite differently from KP⁠. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the axiom of constructibility to KP(P) gives rise to a much stronger theory, whereas in the case of KP⁠, the constructible hierarchy provides an inner model, so that KP and KP+V=L have the same strength. This paper will be concerned with the relationship between KP(P) and KP(P) plus the axiom of choice or even the global axiom of choice, ACglobal⁠. Since L is the standard vehicle to furnish a model in which this axiom holds, the usual argument for demonstrating that the addition of AC or ACglobal to KP(P) does not increase proof-theoretic strength does not apply in any obvious way. Among other tools, the paper uses techniques from ordinal analysis to show that KP(P)+ACglobal has the same strength as KP(P)⁠, thereby answering a question of Mathias. Moreover, it is shown that KP(P)+ACglobal is conservative over KP(P) for Π14 statements of analysis. The method of ordinal analysis for theories with power set was developed in an earlier paper. The technique allows one to compute witnessing information from infinitary proofs, providing bounds for the transfinite iterations of the power set operation that are provable in a theory. As the theory KP(P)+ACglobal provides a very useful tool for defining models and realizability models of other theories that are hard to construct without access to a uniform selection mechanism, it is desirable to determine its exact proof-theoretic strength. This knowledge can for instance be used to determine the strength of Feferman’s operational set theory with power set operation as well as constructive Zermelo–Fraenkel set theory with the axiom of choice

The scope of Feferman’s semi-intuitionistic set theories and his second conjecture

The paper is concerned with the scope of semi-intuitionistic set theories that relate to various foundational stances. It also provides a proof for a second conjecture of Feferman’s that relates the concepts for which the law of excluded middle obtains to those that are absolute with regard to the relevant test structures, or more precisely of ∆1 complexity. The latter is then used to show that a plethora of statements is indeterminate with respect to various semi-intuitionistic set theories

The Use of Trustworthy Principles in a Revised Hilbert’s Program

After the failure of Hilbert’s original program due to Gödel’s second incompleteness theorem, relativized Hilbert’s programs have been sug-gested. While most metamathematical investigations are focused on car-rying out mathematical reductions, we claim that in order to give a full substitute for Hilbert’s program, one should not stop with purely mathe-matical investigations, but give an answer to the question why one should believe that all theorems proved in certain mathematical theories are valid. We suggest that, while it is not possible to obtain absolute certainty, it is possible to develop trustworthy core principles using which one can prove the correctness of mathematical theories. Trust can be established by both providing a direct validation of such principles, which is nec-essarily non-mathematical and philosophical in nature, and at the same time testing those principles using metamathematical investigations. We investigate three approaches for trustworthy principles, namely ordinal no-tation systems built from below, Martin-Löf type theory, and Feferman’s system of explicit mathematics. We will review what is known about the strength up to which direct validation can be provided. 1 Reducing Theories to Trustworthy Principles In the early 1920’s Hilbert suggested a program for the foundation of mathemat-ics, which is now called Hilbert’s program. As formulated in [40], “it calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof it-self was to be carried out using only what Hilbert called ’finitary ’ methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics. ” Because of Gödel’s second incomplete-ness theorem, Hilbert’s program can be carried out only for very weak theories

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

Ordinal analysis and the set existence property for intuitionistic set theories.

On account of being governed by constructive logic, intuitionistic theories T often enjoy various existence properties. The most common is the numerical existence property (NEP). It entails that an existential theorem of T of the form (∃x∈N)A(x) can be witnessed by a numeral n¯ such that T proves A(n¯). While NEP holds almost universally for natural intuitionistic set theories, the general existence property (EP), i.e. the property of a theory that for every existential theorem, a provably definable witness can be found, is known to fail for some prominent intuitionistic set theories such as Intuitionistic Zermelo–Fraenkel set theory (IZF) and constructive Zermelo–Fraenkel set theory (CZF). Both of these theories are formalized with collection rather than replacement as the latter is often difficult to apply in an intuitionistic context because of the uniqueness requirement. In light of this, one is clearly tempted to single out collection as the culprit that stymies the EP in such theories. Beeson stated the following open problem: ‘Does any reasonable set theory with collection have the existence property? and added in proof: The problem is still open for IZF with only bounded separation.’ (Beeson. 1985 Foundations of constructive mathematics, p. 203. Berlin, Germany: Springer.) In this article, it is shown that IZF with bounded separation, that is, separation for formulas in which only bounded quantifiers of the forms (∀x∈a),(∃x∈a),(∀x⊆a),(∃x⊆a) are allowed, indeed has the EP. Moreover, it is also shown that CZF with the exponentiation axiom in place of the subset collection axiom has the EP. Crucially, in both cases, the proof involves a detour through ordinal analyses of infinitary systems of intuitionistic set theory, i.e. advanced techniques from proof theory

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

Naming the largest number: Exploring the boundary between mathematics and the philosophy of mathematics

What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the spirit of the problem, and those that turn it back into a philosophical problem