arXiv subjects

Michael Rathjen

Publications and source records attributed to Michael Rathjen.

At least 19 recordsLinked to original sources

arXiv2024

Choice and independence of premise rules in intuitionistic set theory

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and G\"odel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over $\mathbb{N}$. It is also shown that the existence property (or existential definability property) holds for statements of the form $\exists y^{\sigma}\, \varphi(y)$, where the variable $y$ ranges over objects of finite type $\sigma$. This applies in particular to ${\sf CZF}$ (Constructive Zermelo-Fraenkel set theory) and ${\sf IZF}$ (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.

math.LO↗

arXiv2024

Feferman's completeness theorem

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman's completeness theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.

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arXiv2023

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.

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arXiv2023

Admissible extensions of subtheories of second order arithmetic

In this paper we study admissible extensions of several theories T of reverse mathematics. The idea is that in such an extension the structure M = (N,S,\in) of the natural numbers N and collection of sets of natural numbers S has to obey the axioms of T while simultaneously one also has a set-theoretic world with transfinite levels erected on top of M governed by the axioms of Kripke-Platek set theory, KP.

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arXiv2022

Kreisel-Lévy-type theorems for Kripke-Platek and other set theories

We prove that, over Kripke-Platek set theory with infinity (KP), transfinite induction along the ordinal $ε_{Ω+1}$ is equivalent to the schema asserting the soundness of KP, where $Ω$ denotes the supremum of all ordinals in the universe; this is analogous to the result that, over Peano arithmetic (PA), transfinite induction along $ε_0$ is equivalent to the schema asserting the soundness of PA. In the proof we need to code infinitary proofs within KP, and it is done by using partial recursive set functions. This result can be generalised to KP + $Γ$-separation + $Γ$-collection where $Γ$ is any given syntactic complexity, but not to ZF.

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arXiv2022

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

In this work we consider an extension MFcind of the Minimalist Foundation MF for predicative constructive mathematics with the addition of inductive and coinductive definitions sufficient to generate Sambin's Positive topologies, namely Martin-Löf-Sambin formal topologies equipped with a Positivity relation (used to describe pointfree formal closed subsets). In particular the intensional level of MFcind, called mTTcind, is defined by extending with coinductive definitions another theory mTTind extending the intensional level mTT of MF with the sole addition of inductive definitions. In previous work we have shown that mTTind is consistent with Formal Church's Thesis CT and the Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show the expectation that the addition of coinductive definitions to mTTind does not increase its consistency strength by reducing the consistency of mTTcind+CT+AC to the consistency of CZF+REA through various interpretations. We actually reach our goal in two ways. One way consists in first interpreting mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of Martin-Löf's type theory with Palmgren's superuniverse S. A last step consists in interpreting MLS* back into CZF+REA. The alternative way consists in first interpreting mTTcind+AC+CT directly in a version of Martin-Löf's type theory with Palmgren's superuniverse extended with CT, which is then interpreted back to CZF+REA. A key benefit of the first way is that the theory CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the extensional level of MFcind. Finally, all the theories considered, except mTTcind+AC+CT, are shown to be of the same proof-theoretic strength.

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arXiv2021

Well ordering principles for iterated $Π^1_1$-comprehension

We introduce ordinal collapsing principles that are inspired by proof theory but have a set theoretic flavor. These principles are shown to be equivalent to iterated $Π^1_1$-comprehension and the existence of admissible sets, over weak base theories. Our work extends a previous result on the non-iterated case, which had been conjectured in Montalbán's "Open questions in reverse mathematics" (Bull. Symb. Log. 17(3)2011). This previous result has already been applied to the reverse mathematics of combinatorial and set theoretic principles. The present paper is a significant contribution to a general approach that connects these fields.

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arXiv2021

Derivatives of normal functions in reverse mathematics

Consider a normal function $f$ on the ordinals (i. e. a function $f$ that is strictly increasing and continuous at limit stages). By enumerating the fixed points of $f$ we obtain a faster normal function $f'$, called the derivative of $f$. The present paper investigates this important construction from the viewpoint of reverse mathematics. Within this framework we must restrict our attention to normal functions $f:\aleph_1\rightarrow\aleph_1$ that are represented by dilators (i. e. particularly uniform endofunctors on the category of well-orders, as introduced by J.-Y. Girard). Due to a categorical construction of P. Aczel, each normal dilator $T$ has a derivative $\partial T$. We will give a new construction of the derivative, which shows that the existence and fundamental properties of $\partial T$ can already be established in the theory $\mathbf{RCA}_0$. The latter does not prove, however, that $\partial T$ preserves well-foundedness. Our main result shows that the statement ``for every normal dilator $T$, its derivative $\partial T$ preserves well-foundedness'' is $\mathbf{ACA}_0$-provably equivalent to $Π^1_1$-bar induction (and hence to $Σ^1_1$-dependent choice and to $Π^1_2$-reflection for $ω$-models).

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arXiv2021

A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice

We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short mtt, extended with inductively generated formal topologies, Church's thesis and axiom of choice. This semantics is an extension of the one used to show consistency of the intensional level of the Minimalist Foundation with the axiom of choice and formal Church's thesis in previous work. A main novelty here is that such a semantics is formalized in a constructive theory represented by Aczel's constructive set theory CZF extended with the regular extension axiom.

math.LO↗

arXiv2021

No speedup for geometric theories

Geometric theories based on classical logic are conservative over their intuitionistic counterparts for geometric implications. The latter result (sometimes referred to as Barr's theorem) is squarely a consequence of Gentzen's Hauptsatz. Prima facie though, cut elimination can result in superexponentially longer proofs. In this paper it is shown that the transformation of a classical proof of a geometric implication in a geometric theory into an intuitionistic proof can be achieved in feasibly many steps.

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arXiv2020

Extensional realizability for intuitionistic set theory

In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe $\mathrm{V_{ex}}(A)$ in which the axiom of choice in all finite types ${\sf AC}_{\sf FT}$ is realized, where $A$ stands for an arbitrary partial combinatory algebra. This construction furnishes 'inner models' of many set theories that additionally validate ${\sf AC}_{\sf FT}$, in particular it provides a self-validating semantics for $\sf CZF$ (Constructive Zermelo-Fraenkel set theory) and $\sf IZF$ (Intuitionistic Zermelo-Fraenkel set theory). One can also add large set axioms and many other principles.

math.LO↗

arXiv2020

Ackermann and Goodstein go functorial

We present variants of Goodstein's theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that they (necessarily) entail the existence of complex infinite objects. As part of our proof, we show that the Veblen hierarchy of normal functions on the ordinals is closely related to an extension of the Ackermann function by direct limits.

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arXiv2020

Well-Ordering Principles in Proof Theory and Reverse Mathematics

Several theorems about the equivalence of familiar theories of reverse mathematics with certain well-ordering principles have been proved by recursion-theoretic and combinatorial methods (Friedman, Marcone, Montalban et al.) and with far-reaching results by proof-theoretic technology (Afshari, Freund, Girard, Rathjen, Thomson, Valencia Vizcano, Weiermann et al.), employing deduction search trees and cut elimination theorems in infinitary logics with ordinal bounds in the latter case. At type level one, the well-ordering principles are of the form (*) "if X is well-ordered then f(X) is well-ordered" where f is a standard proof theoretic function from ordinals to ordinals (such f's are always dilators). One aim of the paper is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from (*) and even beta-models from the type 2 version of (*). As (*) is of complexity Pi-1-2 such a principle cannot characterize stronger comprehensions at the level of Pi-1-1-comprehension. This requires a higher order version of (*) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory. The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to Pi-1-1-comprehension. This result has been conjectured and a proof strategy adumbrated roughly 10 years ago, but the proof has only been worked out in recent years.

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arXiv2020

Minimal bad sequences are necessary for a uniform Kruskal theorem

The minimal bad sequence argument due to Nash-Williams is a powerful tool in combinatorics with important implications for theoretical computer science. In particular, it yields a very elegant proof of Kruskal's theorem. At the same time, it is known that Kruskal's theorem does not require the full strength of the minimal bad sequence argument. This claim can be made precise in the framework of reverse mathematics, where the existence of minimal bad sequences is equivalent to a principle known as $Π^1_1$-comprehension, which is much stronger than Kruskal's theorem. In the present paper we give a uniform version of Kruskal's theorem by relativizing it to certain transformations of well partial orders. We show that $Π^1_1$-comprehension is equivalent to our uniform Kruskal theorem (over $\mathbf{RCA}_0$ together with the chain-antichain principle). This means that any proof of the uniform Kruskal theorem must entail the existence of minimal bad sequences. As a by-product of our investigation, we obtain uniform proofs of several Kruskal-type independence results.

math.LO↗

arXiv2019

The independence of premise rule in intuitionistic set theories

Independence of premise principles play an important role in characterizing the modified realizability and the Dialectica interpretations. In this paper we show that a great many intuitionistic set theories are closed under the corresponding independence of premise rule for finite types over $\mathbb{N}$. It is also shown that the existence property (or existential definability property) holds for statements of the form $\neg A\to \exists x^σ F(x^σ)$, where the variable $x^σ$ ranges over a finite type $σ$. This applies in particular to Constructive Zermelo-Fraenkel Set Theory (CZF) and Intuitionistic Zermelo-Fraenkel Set Theory (IZF), two systems known not to have the general existence property. On the technical side, the paper uses the method of realizability with truth from [21] and [8] with the underlying partial combinatory algebra (pca) chosen among the total ones. A particular instance of the latter is provided by the substructure of the graph model formed by the semi computable subsets of $\mathbb{N}$, which has the advantage that it forms a set pca even in proof-theoretically weak set theories such as CZF.

math.LO↗

arXiv2019

Upper bounds on the graph minor theorem

Lower bounds on the proof-theoretic strength of the graph minor theorem were found over 30 years ago by Friedman, Robertson and Seymour 1987, but upper bounds have always been elusive. We present recently found upper bounds on the graph minor theorem and other theorems appearing in the Graph Minors series. Further, we give some ideas as to how the lower bounds on some of these theorems might be improved.

math.LO↗

arXiv2018

Lifschitz Realizability as a Topological Construction

We develop a number of variants of Lifschitz realizability for CZF by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of LLPO including consistency with unique Church's thesis, consistency with some Brouwerian principles and variants of the numerical existence property.

math.LO↗

arXiv2018

Proof Theory of Constructive Systems: Inductive Types and Univalence

In Feferman's work, explicit mathematics and theories of generalized inductive definitions play a central role. One objective of this article is to describe the connections with Martin-Lof type theory and constructive Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of the relationship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky's univalence axiom. A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin-Lof type theory.

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