This paper is dedicated to Prof. Dr. Günter Asser, whose work in founding this journal and maintaining it over many difficult years has been a major contribution to the activities of the mathematical logic community.At first sight it appears highly unlikely that Urysohn's Lemma has any significant constructive content. However, working in the context of an apartness space and using functions whose values are a generalisation of the reals, rather than real numbers, enables us to produce a significant (...) constructive version of that lemma. (shrink)

We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.

Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying V=HOD. We show that the lemma fails in the forcing extension of the universe by (...) adding a Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht’s lemma, namely, the principles of the form EL, which asserts that P-definability from A implies Q-discernibility. We also consider various analogues of Ehrenfeucht’s lemma obtained by using algebraicity in place of definability, where a set b is algebraic in a if it is a member of a finite set definable from a. Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using a similar analysis, we answer two open questions posed earlier by the third author and C. Leahy, showing that algebraicity and definability need not coincide in models of set theory and the internal and external notions of being ordinal algebraic need not coincide. (shrink)

We explore the connections between Dickson’s lemma and weak Ramsey theory. We show that a weak version of the Paris–Harrington principle for pairs in c colors and miniaturized Dickson’s lemma for c-tuples are equivalent over \. Furthermore, we look at a cascade of consequences for several variants of weak Ramsey’s theorem.

Bishop's Lemma is a centrepiece in the development of constructive analysis. We show that its proof requires some form of the axiom of choice; and that the completeness requirement in Bishop's Lemma can be weakened and that there is a vast class of non‐complete spaces that Bishop's Lemma applies to.

We provide a fine‐grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable (...) fan theorem is not equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation. (shrink)

Classically, weak König's lemma and Brouwer's fan theorem for detachable bars are equivalent. We give a direct constructive proof that the former implies the latter.

The exponential rise in transgender self-identification invites consideration of what constitutes an ethical response to transgender individuals’ claims about how best to promote their well-being. In this paper, we argue that ‘accepting’ a claim to medical transitioning in order to promote well-being would be in the person’s best interests iff at the point of request the individual is correct in their self-diagnosis as transgender (i.e., the distress felt to reside in the body does not result from another psychological and/or societal (...) problem) such that the medical interventions they are seeking will help them to realise their preferences. If we cannot assume this—and we suggest that we have reasonable grounds to question an unqualified acceptance in some cases—then ‘acceptance’ potentially works against best interests. We propose a distinction between ‘acceptance’ and respectful, in-depth exploration of an individual’s claims about what promotes their well-being. We discuss the ethical relevance of the unconscious mind to considerations of autonomy and consent in working with transgender individuals. An inquisitive stance, we suggest, supports autonomous choice about how to realise an embodied form that sustains well-being by allowing the individual to consider both conscious and unconscious factors shaping wishes and values, hence choices. (shrink)

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence (...) of some familiar results in the theory of (complete) Boolean algebras--notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent. (shrink)

Notini et al 1 offer a timely addition in the wake of a significant increase in young people identifying as transgender and gender diverse. The authors focus specifically on the case of 18-year-old Phoenix’s request for ongoing puberty suppression to affirm a non-binary gender identity. A central issue raised by Phoenix’s predicament, and that I suggest we can extend to ethical consideration of requests for other types of medical intervention by binary and non-binary TGD individuals, is whether we should ‘affirm’ (...) the claims made by TGD young people about what promotes their well-being. As a clinician working with TGD young people who have undergone a range of medical interventions to align their bodies with their gender identity, my starting presumption is that it is important to respect what people think will help them: there are some TGD young people for whom, all things considered, medical interventions to affirm gender identity promote their well-being. Well-being has become an important conceptual currency in ethical debates and lies at the core of the authors’ analysis of Phoenix’s case. While the authors do not ground their argument in favour of OPS in a specific theory of well-being, welfarist considerations 2 appear to underpin …. (shrink)

The deductive strengths of three variations of Rado's selection lemma are studied in set theory without the axiom of choice. Two are shown to be equivalent to Rado's lemma and the third to the Boolean prime ideal theorem. MSC: 03E25, 04A25, 06E05.

q-derivatives are part of so called quantum calculus. In this paper we investigate how such derivatives can possibly be used in Itô’s lemma. This leads us to consider how such derivatives can be used in a social science setting. We conclude that in a Itô Lemma setting we cannot use a macroscopic version of the Heisenberg uncertainty principle with q-derivatives.

We show how to interpret weak Kőnig's lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Friedman's about weak Kőnig's lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in herbrandized functional interpretations.

Rauszer and Sabalski proved in [2] that distributivity with respect to infi- nite joins and meets is a sucient and necessary condition making the RasiowaSikorski Lemma valid in distributive lattices. The main part of their proof is a direct construction of a required filter under distributivity. In this note we show that a generalization of the result can be obtained from the Rasiowa-Sikorski Lemma for Boolean algebras by using Gornemann’s result in [1] instead of a direct con- ¨ (...) struction. Suppose A is a distributive lattice and Q; R 2 A f;g. We call A complete if 8M 2 Q9 M 2 A and 8N 2 R9 F N 2 A. M 2 Q is -dis if 9 M 2 A and 8a 2 A a t M = m2M. N 2 R is F -dis if 9 F N 2 A and 8a 2 A a u F N = F n2N. is called distributive if every M 2 Q is -dis and every N 2 R is F -dis. Suppose A is complete and C; D A. By rC we mean the filter in A generated by C, in particular, let r; = ; = ;. is called complete if and , where 8M 2 Q 9m 2 MrC \ D[fmg = ;) 8N 2 R 9n 2 NrC[fng \ D = ;). (shrink)

The so-called weak König's lemma WKL asserts the existence of an infinite path b in any infinite binary tree . Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are Π 2 0 -conservative over primitive recursive arithmetic PRA . In Kohlenbach 1239–1273) we established such conservation results relative to finite type extensions PRA ω of PRA . In this setting one can consider also a (...) uniform version UWKL of WKL which asserts the existence of a functional Φ which selects uniformly in a given infinite binary tree f an infinite path Φf of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in Kohlenbach [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA ω only has a quantifier-free rule of extensionality QF-ER instead of the full axioms of extensionality for all finite types. In this paper we show that in the presence of , UWKL is much stronger than WKL: whereas WKL remains to be Π 2 0 -conservative over PRA, PRA ω ++ UWKL contains full Peano arithmetic PA. We also investigate the proof–theoretic as well as the computational strength of UWKL relative to the intuitionistic variant of PRA ω both with and without the Markov principle. (shrink)

In this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

Two new intuitionistic equivalents to Zarn’s Lemma are stated and proved.

J. S. Bell's classic 1966 review paper on the foundations of quantum mechanics led directly to the Bell nonlocality theorem. It is not widely appreciated that the review paper contained the basic ingredients needed for a nonlocality result which holds in certain situations where the Bell inequality is not violated. We present in this paper a systematic formulation and evaluation of an argument due to Stairs in 1983, which establishes a nonlocality result based on the Bell-Kochen-Specker “paradox” in quantum mechanics.

A general algorithm is given for determining whether or not a given set of pair distributions allows for the construction of all the members of a specified set of higher-order distributions which return the given pair distributions as marginals. This mathematical question underlies studies of quantum correlation experiments such as those of Bell or of Clauser and Horne, or their higher-spin generalizations. The algorithm permits the analysis of rather intricate versions of such problems, in a form readily adaptable to the (...) computer. The general procedure is illustrated by simple derivations of the results of Mermin and Schwarz for the symmetric spin-1 and spin-3/2 Einstein-Podolsky-Rosen problems. It is also used to extend those results to the spin-2 and spin-5/2 cases, providing further evidence that the range of strange quantum theoretic correlations does not diminish with increasing s. The algorithm is also illustrated by giving an alternative derivation of some recent results on the necessity and sufficiency of the Clauser-Horne conditions. The mathematical formulation of the algorithm is given in general terms without specific reference to the quantum theoretic applications. (shrink)

I derive a Gleason-type contradiction from assumptions weaker than those needed to reach a Bell inequality. By establishing the inconsistency between local realism and QM's perfect EPR-type anticorrelations, the proof fills in a gap left open by Bell arguments.

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.

We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is (...) countably additive on open sets. (shrink)

A family \ is called Rosenthal if for every Boolean algebra \, bounded sequence \ of measures on \, antichain \ in \, and \, there exists \ such that \<\varepsilon \) for every \. Well-known and important Rosenthal’s lemma states that \ is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in \}\) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly (...) less than \\), the covering of category. We also study ultrafilters on \ which are Rosenthal families—we show that the class of Rosenthal ultrafilters contains all selective ultrafilters. (shrink)

We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.

By , we denote the system of second-order arithmetic based on recursive comprehension axioms and Σ10 induction. is defined to be plus weak König's lemma: every infinite tree of sequences of 0's and 1's has an infinite path. In this paper, we first show that for any countable model M of , there exists a countable model M′ of whose first-order part is the same as that of M, and whose second-order part consists of the M-recursive sets and sets (...) not in the second-order part of M. By combining this fact with a certain forcing argument over universal trees, we obtain the following result : if proves X!Y with arithmetical, so does . We also discuss several improvements of this results. (shrink)

We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL 0 over RCA 0 . We show that the separation theorem for separably closed convex sets is equivalent to ACA 0 over RCA 0 . Our strategy for proving these geometrical Hahn-Banach theorems is to reduce to the finite-dimensional case (...) by means of a compactness argument. (shrink)

Principia (Book 1, Sect. 6), Newton's Lemma 28 on the algebraic nonintegrability of ovals has had an unusually mixed reception. Beginning in 1691 with Jakob Bernoulli (who accepted the lemma) and Huygens and Leibniz (who rejected it and offered counterexamples), Lemma 28 has a history of eliciting seemingly contradictory reactions. In more recent times, D.T. Whiteside in 1974 gave an “unchallengeable counterexample,” while the mathematician V.I. Arnol'd in 1987 sided with Bernoulli and called Newton's argument an “astonishingly (...) modern topological proof.” This disagreement mostly stems, we argue, from Newton's vague statement of the lemma. Indeed, we identify several different interpretations of Lemma 28, any one of which Newton may have been intending to assert, and we then test a number of proposed counterexamples to see which, if any, are true counterexamples to one or more of these versions of the lemma. Following this, we study Newton's argument for the lemma to see whether and where it fails to be convincing. In the end, our study of Newton's Lemma 28 provides an answer to the question, Who is right: Huygens, Leibniz, Whiteside and the others who reject the lemma, or Bernoulli, Arnol'd, and the others who accept it? (shrink)

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A, B, C such that A (...) ↔, that is, A can be split into two independent parts B and C, and the aforementioned topological notions give rise to a number of splittings involving highly natural A, B, C. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory RCA 0. We obtain similarly “exceptional” results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice. (shrink)

In this paper, we show within RCA 0 that weak Konig's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL 0 , a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL 0 has a proper initial part isomorphic to itself [10].

Every second-countable regular topological space X is metrizable. For a given “computable” topological space satisfying an axiom of computable regularity M. Schröder [10] has constructed a computable metric. In this article we study whether this metric space can be considered computationally as a subspace of some computable metric space [15]. While Schröder's construction is “pointless”, i. e., only sets of a countable base but no concrete points are known, for a computable metric space a concrete dense set of computable points (...) is needed. But there may be no computable points in X. By converging sequences of basis sets instead of Cauchy sequences of points we construct a metric completion of a space together with a canonical representation equation image, equation image. We show that there is a computable embedding of in with computable inverse. Finally, we construct a notation of a dense set of points in with computable mutual distances and prove that the Cauchy representation of the resulting computable metric space is equivalent to equation image. Therefore, every computably regular space has a computable homeomorphic embedding in a computable metric space, which topologically is its completion. By the way we prove a computable Urysohn lemma. (shrink)

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only (...) use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

This is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.

To solve the direct problem of central forces when the trajectory is an ellipse and the force is directed to its centre, Newton made use of the famous Lemma 12 (Principia, I, sect. II) that was later recognized equivalent to proposition 31 of book VII of Apollonius’s Conics. In this paper, in which we look for Newton’s possible sources for Lemma 12, we compare Apollonius’s original proof, as edited by Borelli, with those of other authors, including that given (...) by Newton himself. Moreover, after having retraced its editorial history, we evaluate the dissemination of Borelli's edition of books V-VII of Apollonius’s Conics before the printing of the Principia. (shrink)