We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps (...) are enumeration degree jumps.) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees. (shrink)

We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in e α if and only if there is a Turing machine which computes X and "halts" in less than or equal to the ordinal number ω α of steps. This result represents a generalization of the well-known "limit lemma" due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application (...) of this result, we give a recursion theoretic analysis of clopen determinacy: there is a correlation given between the height α of a well-founded tree corresponding to a clopen game $A \subseteq \omega^\omega$ and the Turing degree of a winning strategy f for one of the players--roughly, f can be chosen to be recursive in 0 α and this is the best possible (see § 4 for precise results). (shrink)

A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent formulation (...) of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of [Formula: see text] for all [Formula: see text]. (shrink)

This paper examines the ramifications of Hume's view of the relation of conceivability to metaphysical possibility. It argues that the limitations Hume places of the representations involved in moves to conceivability to metaphysical possibility preclude any straightforward argument against full-blooded causal realism in Hume from conceivability. Furthermore, our finding certain states of affairs conceivable when they are not metaphysically possible is perfectly compatible with the thrust of the causal realist position.

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)

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)

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... (shrink)

Propositional dynamic logic is complete but not compact. As a consequence, strong completeness requires an infinitary proof system. In this paper, we present a short proof for strong completeness of $$\mathsf{PDL}$$ relative to an infinitary proof system containing the rule from [α; β n ]φ for all $$n \in {\mathbb{N}}$$, conclude $$[\alpha;\beta^*] \varphi$$. The proof uses a universal canonical model, and it is generalized to other modal logics with infinitary proof rules, such as epistemic knowledge with common knowledge. Also, we (...) show that the universal canonical model of $$\mathsf{PDL}$$ lacks the property of modal harmony, the analogue of the Truth lemma for modal operators. (shrink)

Our account of the problem of the classical limit of quantum mechanics involves two elements. The first one is self-induced decoherence, conceived as a process that depends on the own dynamics of a closed quantum system governed by a Hamiltonian with continuous spectrum; the study of decoherence is addressed by means of a formalism used to give meaning to the van Hove states with diagonal singularities. The second element is macroscopicity represented by the limit $\hbar \rightarrow 0$ : (...) when the macroscopic limit is applied to the Wigner transformation of the diagonal state resulting from decoherence, the description of the quantum system becomes equivalent to the description of an ensemble of classical trajectories on phase space weighted by their corresponding probabilities. (shrink)

_Alessandra Lemma - Winner of the Levy-Goldfarb Award for Child Psychoanalysis!_ The contemporary relevance of psychoanalysis is being increasingly questioned; _Off the Couch_ challenges this view, demonstrating that psychoanalytic thinking and its applications are both innovative and relevant, in particular to the management and treatment of more disturbed and difficult to engage patient groups. Chapters address: clinical applications in diverse settings across the age range the relevance of psychoanalytic thinking to the practice of CBT, psychosomatics and general psychiatry the (...) contribution of psychoanalytic thinking to mental health policy and the politics of conflict and mediation. This book suggests that psychoanalysis has a vital position within the public health sector and discusses how it can be better utilised in the treatment of a range of mental health problems. It also highlights the role of empirical research in providing a robust evidence base. _Off the Couch_ will be essential reading for those practicing in the field of mental health and will also be useful for anyone involved in the development of mental health and public policies. It will ensure that practitioners and supervisors have a clear insight into how psychoanalysis can be applied in general healthcare. (shrink)

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher (...) random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma. (shrink)

We refine results of Gannon [6, Theorem 4.7] and Simon [22, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual $NIP$, as a property of a model, and prove a variant of [15, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [17] on the model-theoretic meaning of Grothendieck’s double limit theorem.

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The (...) importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. As crucial corner points of this classification scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn—Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof, based on a computational version of Kleene's ternary logic, that the class of weakly computable operations is closed under composition. Moreover, weakly computable operations on computable metric spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it is proved that any single-valued weakly computable operation is already computable in the ordinary sense. (shrink)

An infinite binary sequence X is Kolmogorov–Loveland random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence.One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as KL-randomness. Our (...) first main result states that KL-random sequences are close to Martin-Löf random sequences in so far as every KL-random sequence has arbitrarily dense subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. However, this splitting property does not characterize KL-randomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2-random. Furthermore, we show for any KL-random sequence A that is computable in the halting problem that, first, for any effective split of A both halves are Martin-Löf random and, second, for any computable, nondecreasing, and unbounded function g and almost all n, the prefix of A of length n has prefix-free Kolmogorov complexity at least n−g. Again, the latter property does not characterize KL-randomness, even when restricted to left-r.e. sequences; we construct a left-r.e. sequence that has this property but is not KL-stochastic and, in fact, is not even Mises–Wald–Church stochastic.Turning our attention to KL-stochasticity, we construct a non-empty class of KL-stochastic sequences that are not weakly 1-random; by the usual basis theorems we obtain such sequences that in addition are left-r.e., are low, or are of hyperimmune-free degree.Our second main result asserts that every KL-stochastic sequence has effective dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α<1. This improves on a result by Muchnik, who has shown that were they to exist, such compressible prefixes could not be found effectively. (shrink)

We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2κ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf > ω then it follows that o λ, and if cf = ωthen either o λ or {α: K o α+n} is confinal in κ (...) for each n ε ω.We also prove several results which extend or are related to this result, notably Theorem. If 2ω ω1 then there is a sharp for a model with a strong cardinal.In order to prove these theorems we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders. (shrink)

Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. (...) Furthermore, we give a short and elementary proof of the fact that FAN is equivalent to each positive valued function with compact domain having positive infimum. (shrink)

We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of (...) the new interpretation to theories of classical arithmetic and analysis. (shrink)

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than (...) or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes. (shrink)

A generalization of Příkrý's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkrý generic sequences reminiscent of Mathias' criterion for Příkrý genericity is provided, together with a maximality theorem which states that a generalized Příkrý sequence almost contains every other one lying in the same extension.This forcing can be used to falsify the covering lemma for (...) a higher core model if there is an inner model with infinitely many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov's principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König's lemma. We discuss some applications, as well as some previous results in the literature.

The translation of Newton’s geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which (...) was a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem. (shrink)

We show that the predicate “x is the power set of y” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has (...) a cardinal strong up to one of its ℵ‐fixed points, and, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ. (shrink)

In the 1687 Principia, Newton gave a solution to the direct problem (given the orbit and center of force, find the central force) for a conic-section with a focal center of force (answer: a reciprocal square force) and for a spiral orbit with a polar center of force (answer: a reciprocal cube force). He did not, however, give solutions for the two corresponding inverse problems (given the force and center of force, find the orbit). He gave a cryptic solution to (...) the inverse problem of a reciprocal cube force, but offered no solution for the reciprocal square force. Some take this omission as an indication that Newton could not solve the reciprocal square, for, they ask, why else would he not select this important problem? Others claim that ``it is child's play'' for him, as evidenced by his 1671 catalogue of quadratures (tables of integrals). The answer to that question is obscured for all who attempt to work through Newton's published solution of the reciprocal cube force because it is done in the synthetic geometric style of the 1687 Principia rather than in the analytic algebraic style that Newton employed until 1671. In response to a request from David Gregory in 1694, however, Newton produced an analytic version of the body of the proof, but one which still had a geometric conclusion. Newton's charge is to find both ``the orbit'' and ``the time in orbit.'' In the determination of the dependence of the time on orbital position, t(r), Newton evaluated an integral of the form ∫dx/xn to calculate a finite algebraic equation for the area swept out as a function of the radius, but he did not write out the analytic expression for time t = t(r), even though he knew that the time t is proportional to that area. In the determination of the orbit, θ (r), Newton obtained an integral of the form ∫dx/√(1−x2) for the area that is proportional to the angle θ, an integral he had shown in his 1669 On Analysis by Infinite Equations to be equal to the arcsin(x). Since the solution must therefore contain a transcendental function, he knew that a finite algebraic solution for θ=θ(r) did not exist for ``the orbit'' as it had for ``the time in orbit.'' In contrast to these two solutions for the inverse cube force, however, it is not possible in the inverse square solution to generate a finite algebraic expression for either ``the orbit'' or ``the time in orbit.'' In fact, in Lemma 28, Newton offers a demonstration that the area of an ellipse cannot be given by a finite equation. I claim that the limitation of Lemma 28 forces Newton to reject the inverse square force as an example and to choose instead the reciprocal cube force as his example in Proposition 41. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's (...) class='Hi'>lemma, in the original classical setting of reverse mathematics started by Friedman and Simpson. (shrink)

The concept of elementary epimorphism is introduced. Inverse systems of such maps are considered, and a dual of the elementary chain lemma is found (Cor. 4.2). The same is done for pure epimorphisms (Cor. 4.3 and 4.4). Finally, this is applied to certain inverse limits of flat modules (Thm. 6.4) and certain inverse limits of absolutely pure modules (Cor. 6.3).

The paper presents a generalization of pregroup, by which a freely-generated pregroup is augmented with a finite set of commuting inequations, allowing limited commutativity and cancelability. It is shown that grammars based on the commutation-augmented pregroups generate mildly context-sensitive languages. A version of Lambek’s switching lemma is established for these pregroups. Polynomial parsability and semilinearity are shown for languages generated by these grammars.

We develop an untyped framework for the multiverse of set theory. $\mathsf {ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf {Uni}(\mathcal {U})$ and $\mathsf {Mod}(\mathcal {U, \sigma })$, expressing that $\mathcal {U}$ is a universe and that $\sigma $ is true in the universe $\mathcal {U}$, respectively. Here $\sigma $ ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests its (...) own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf {ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins’ multiverse theory. (shrink)

Transformer is a neural machine translation model which revolutionizes machine translation. Compared with traditional statistical machine translation models and other neural machine translation models, the recently proposed transformer model radically and fundamentally changes machine translation with its self-attention and cross-attention mechanisms. These mechanisms effectively model token alignments between source and target sentences. It has been reported that the transformer model provides accurate posterior alignments. In this work, we empirically prove the reverse effect, showing that prior alignments help transformer models produce (...) better translations. Experiment results on Vietnamese-English news translation task show not only the positive effect of manually annotated alignments on transformer models but also the surprising outperformance of statistically constructed alignments reinforced with the flexibility of token-type selection over manual alignments in improving transformer models. Statistically constructed word-to-lemma alignments are used to train a word-to-word transformer model. The novel hybrid transformer model improves the baseline transformer model and transformer model trained with manual alignments by 2.53 and 0.79 BLEU, respectively. In addition to BLEU score, we make limited human judgment on translation results. Strong correlation between human and machine judgment confirms our findings. (shrink)

For a centerless group G, we can define its automorphism tower. We define G α : G 0 = G, G α+1 = Aut(G α ) and for limit ordinals ${G^{\delta}=\bigcup_{\alpha<\delta}G^{\alpha}}$ . Let τ G be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says ${\tau_{G}<(2^{|G|})^{+}}$ and more. If we consider Thomas’ proof too set theoretical (using Fodor’s lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem (...) without the Axiom of Choice. Moreover, we give a descriptive set theoretic approach for calculating an upper bound for τ G for all countable groups G (better than the one an analysis of Thomas’ proof gives). We attach to every element in G α , the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from ${G*\langle x\rangle}$ ). This situation is generalized by defining “(G, A) is a special pair”. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

This is a remarkable new French language introduction to elementary logical methods. Although designed primarily for computer and information specialists, it is also sure to interest philosophers and logicians because of its diversity of subjects, emphasis on graphic calculation techniques, and extensive historical background. The book is intelligently divided into nine main chapters with detailed descriptive subsections. It begins with the most fundamental principles of Aristotelian syllogistic and Boolean algebra, working through the essentials of Frege's predicate calculus and Gödel's consistency-completeness (...) metaproofs before systematically articulating standard propositional and predicate logics, interestingly, from synthetic and then analytic perspectives. This leads to a largely informal exposition of Tarski-style model set theoretical semantics, and the modern limiting metatheoretical results underlying what the authors call the "crisis" in the foundations of mathematical logic. Here are included such typical and more unusual topics as Herbrand models, König's Lemma, Herbrand's Theorem, Post machines, Floyd's Theorem, and Church's Theorem. The volume culminates in an expansive treatment of first order theories, including theories of order and models, and decidability. The exposition is somewhat rigorous, but should not be excessively demanding for nonprofessionals interested in the important implications of logic for philosophy, foundations of mathematics, and informatique. Among the book's virtues must be mentioned its profuse illustration of so many different graphic methods in mathematical logic, including Venn diagrams, the square of opposition, Smullyan truth trees, Wallen's reduction technique, Beth's and Hintikka's semantic tableaux method, and several styles of axiomatic and natural deduction. The work is certain to be a valuable resource for French readers in need of a lucid, comprehensive guide to the history and practical techniques of logic in its elementary applications to the philosophy of mathematics and information sciences.--Dale Jacquette, The Pennsylvania State University. (shrink)

Lexicalized theories of syntax often assume that verb-structure regularities are mediated by lemmas, which abstract over variation in verb tense and aspect. German syntax seems to challenge this assumption, because verb position depends on tense and aspect. To examine how German speakers link these elements, a structural priming study was performed which varied syntactic structure, verb position, and verb overlap.structural priming was found, both within and across verb position, but priming was larger when the verb position was the same between (...) prime and target. Priming was boosted by verb overlap, but there was no interaction with verb position. The results can be explained by a lemma model where tense and aspect are linked to structural choices in German. Since the architecture of this lemma model is not consistent with results from English, a connectionist model was developed which could explain the cross-linguistic variation in the production system. Together, these findings support the view that language learning plays an important role in determining the nature of structural priming in different languages. (shrink)

Jensen's celebrated Covering Lemma states that if 0# does not exist, then for any uncountable set of ordinals X, there is a Y∈L such that X⊆Y and |X| = |Y|. Working in ZF + AD alone, we establish the following analog: If ℝ# does not exist, then L(ℝ) and V have exactly the same sets of reals and for any set of ordinals X with |X| ≥ΘL(ℝ), there is a Y∈L(ℝ) such that X⊆Y and |X| = |Y|. Here ℝ (...) is the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ. (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.

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)