We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy (...) [Formula: see text][Formula: see text] or under [Formula: see text] there are no infinite [Formula: see text]-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal [Formula: see text], which corresponds to the classical notion of MAD families, as well as to the ideal [Formula: see text]. The proofs combine ideas from invariant descriptive set theory and forcing. (shrink)

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. (...) The technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy ), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \ from the point of view of the research area in question is its game-theoretic character. (shrink)

We prove within K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K}$$\end{document} that the axiom of determinacy is equivalent to the assertion that for each ordinal λ λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa > \lambda}$$\end{document}. Here Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta}$$\end{document} is the supremum of the ordinals which are the surjective image of the set of reals R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}.

Two-player, zero-sum, non-cooperative, blindfold games in extensive form with incomplete information are considered in this paper. Any information about past moves which players played is stored in a database, and each player can access the database. A polynomial game is a game in which, at each step, all players withdraw at most a polynomial amount of previous information from the database. We show resource-bounded determinacy of some kinds of finite, zero-sum, polynomial games whose pay-off sets are computable by non-deterministic (...) polynomial-time function-oracle Turing machines. We call a pay-off set -determined if, for any polynomial game G associated with the given pay-off set, either player has a winning strategy which is in for any subgames of G. We show that there exists an FP-strongly-determined pay-off set which is computed by an exponential-time oracle Turing machine, where FP is the set of polynomial-time computable functions. We also discuss several relationships between the determinacy of polynomial games and recursion-theoretic properties for the classes co-NP and co-NEXP. We show that the polynomial version of the axiom of choice holds under some assumption of polynomial determinacy for a pay-off set which is polynomial-time computable with parallel queries. This principle of choice implies that co-NP has the separation property. (shrink)

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of (...) van Douwen, and gives partial answers to questions of Terada and Medvedev. (shrink)

We define a parametrised choice principle PCP which is equivalent to the Axiom of Determinacy. PCP describes the difference between these two axioms and could serve as a means of proving Martin's conjecture on the equivalence of these axioms.

We show that the Axiom of Dependent Choice,, holds in countably iterable, passive premice constructed over their reals which satisfy the Axiom of Determinacy,, in a background universe. This generalizes an argument of Kechris for using Steel's analysis of scales in mice. In particular, we show that for any and any countable set of reals A so that and, we have that.

The theory of infinite games with slightly imperfect information has been developed for games with finitely and countably many moves. In this paper, we shift the discussion to games with uncountably many possible moves, introducing the axiom of real Blackwell determinacy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{Bl-AD}_\mathbb{R}}$$\end{document} (as an analogue of the axiom of real determinacy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{AD}_\mathbb{R}}$$\end{document}). We prove that the consistency strength of (...) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{Bl-AD}_\mathbb{R}}$$\end{document} is strictly greater than that of AD. (shrink)

Pursues the theoretical level of the two‐tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I outline the pre‐axiomatic development of set theory out of Cantor's researches, describe how axiomatization arose in the course of Zermelo's efforts to prove Cantor's Well‐ordering Theorem, and review the controversy over the Axiom of Choice. Cantor's Continuum Hypothesis and various questions of descriptive set theory were eventually shown to be (...) independent of the standard axioms, and new axiom candidates—Gödel's Axiom of Constructibility, on the one hand, determinacy and large cardinal axioms, on the other—offer dramatically different solutions. This situation presents a new epistemic challenge to the set theoretic realist: on what grounds can we adjudicate between new axiom candidates? (shrink)

In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, (...) but the reversal is provable even in ACA 0 . These results illuminate the substantial differences among lightface theorems which can not be observed in boldface. (shrink)

Assume V = L ⊨ ZF + DC + Θ > ω2 + μ is a normal fine measure on.

Assume ZF + AD +V=L and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof = ω thenκis Rowbottom. We also establish some other partition properties.

The problem of determinacy versus indeterminacy in morals is the problem of the foundation of ethical judgment. The most relative of relativisms, the emotive theory of ethics, has in its most flagrant and simplified form held moral judgments to rest basically upon attitudes and therefore to be neither true nor false. As ethical disagreements involve oppositions of attitudes, they cannot be resolved through any appeal, however attractive to the cognitivist, to beliefs. That ethical disagreement is ultimate simply because individuals (...) differ in their attitudes, is challenged by Professor Edel as a semi-apriorism which may be broken through an appeal, in particular cases, to the steadily increasing knowledge of human life. The removal of arbitrary barriers raised by the emotivist and others opens the way to the recovery of the cognitive element as a basis of moral judgment and the restoration of ethics as a practical science. (shrink)

Many theorists understand human rights as only aiming to secure a minimally decent existence, rather than a positively good or flourishing life. Some of the theoretical considerations that support this minimalist view have been mapped out in the philosophical literature. The aim of this paper is to explain how a relatively neglected theoretical desideratum – namely, determinacy – can be invoked in arguing for human rights minimalism. Most of us want a theory of human rights whose demands can be (...) realized, and which is acceptable to a range of worldviews. But we might also expect our theory to provide determinate answers to questions of scope (i.e. which putative rights are bona fide human rights?) and practical implementation (i.e. what concrete duties are generated by which rights?). A minimalist view of human rights makes it is easier to jointly fulfil all of these desiderata. (shrink)

This paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic D and Propositional Independence Logic I are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics L_D and L_I, based on Kripke semantics, and propose them as alternatives for D and I, respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We (...) argue that L_D and L_I naturally resolve a range of interpretational problems that arise in D and I.We also obtain sound and complete axiomatizations for L_D and L_I . (shrink)

This paper addresses debates in German idealism that arise in response to the modal shift in logic, proposed by Kant, from a logic of thinking to a logic of experience. With the Kantian logic of experience arises a problem of radical contingency or 'rhapsodic determination' for logic. While Fichte and Hegel attempt to resolve the problem of contingency by constructing rational systems aimed at established the grounds for logic, I show how Schelling brings into view, in a proto-existentialist movement, the (...) way in which the indeterminacy of the will undergirds rational system construction and, thereby, presents a limiting problem for the question of the value of system-building itself. (shrink)

The idea that legal theories seek not only to explain but to evaluate the moral justification of particular areas of law is quite familiar. Yet little attention has been paid to the minimal criteria of adequacy for justificatory legal theories. Whereas many theories claim to identify the moral grounds that justify a particular area of law, such as contracts or torts, none of them explains how its justification determines the outcomes of adjudication governed by the law in that area. In (...) this brief Essay for the William and Mary Law Review Symposium on Law and Morality, I argue that a particular area of law can be justified only by identifying moral reasons that fully determine the results of adjudication. No matter how compelling the moral reasons a legal theory identifies, and how tight the fit between those reasons and the structure and content of the legal rules governing a judicial decision, a legal theory fails to justify a particular area of law if the reasoning it identifies falls short of fully determining the results in the judicial decisions governed by that law. Though this bold claim may seem unrealistic, I argue that legal theories can satisfy this determinacy requirement by identifying determinate but inconclusive reasoning that explains outcomes in adjudication. While such reasoning may prove to be erroneous, that does not undermine its justificatory force. (shrink)

Let o(κ) denote the Mitchell order of κ. We show how to reduce long games which run to the first ordinal admissible in the play, to iteration games on models with a cardinal κ so that (1) κ is a limit of Woodin cardinals: and (2) o(κ) = κ⁺⁺. We use the reduction to derive several optimal determinacy results on games which run to the first admissible in the play.

This chapter analyzes the different ways to describe brain behaviour with the goal to provide a basis for an informed discussion of the nature of decisions and actions that humans perform in their lives. The chapter is organized as follows. Section 2 outlines a number of concepts exhibiting how many subtle details and distinctions lie behind the broad notions of determinacy and stochasticity. These details are necessary for a discussion, in Section 3, of particular aspects relevant for the characterization (...) of brain states and their dynamics. The descriptions of brain behaviour currently provided by neuroscience depend on the level and context of the descriptions. There is no clear-cut evidence for ultimately determinate or ultimately stochastic brain behaviour. As a consequence, there is no solid neurobiological basis to argue either in favour of or against any fundamental determination or openness of human decisions and actions. (shrink)

The paper discusses some ways in which the phenomenon of borderline cases may be thought to bear on the traditional philosophical idea that certain domains of facts are fully open to our view. The discussion focusses on a very influential argument (due to Tim Williamson) to the effect that, roughly, no such domains of luminous facts exist. Many commentators have felt that the vagueness unavoidably inherent in the description of the facts that are best candidates for being luminous plays an (...) illicit role in the argument. The paper investigates this issue by centring around the idea that vagueness brings with itself borderline cases, and that these in turn generate absence of a fact of the matter and hence epistemically benign lack of knowledge. It is argued that, given the possibility of absence of a fact of the matter, the idea of luminosity should be reformulated using the notion of determinacy, and that the resulting reformulation is not immediately subject to the original anti-luminosity argument. However, it is shown that the specific understanding of determinacy required by this strategy validates a new argument against the reformulated version of luminosity. Moreover, reflection on the connection between mistake and absence of a fact of the matter offers another argument against such version, with the surprising upshot that, granting the soundness of the original anti-luminosity argument, not even the determinacy of a certain fact would guarantee its knowability. (shrink)

For several partial sharp functions # on the reals, we characterize in terms of determinacy, the existence of indiscernibles for several inner models of “# exists for every real r”. Let #10=1#10 be the identity function on the reals. Inductively define the partial sharp function, β#1γ+1, on the reals so that #1γ+1 =1#1γ+1 codes indiscernibles for L [#11, #12,…, #1γ] and #1γ+1=#1γ+1). We sho w that the existence of β#1γ follows from the determinacy of *Σ01)*+ games . Part (...) I proves the converse. (shrink)

Of the many ways of getting at the core of the weirdnesses in quantum mechanics, there’s one which traces back to Schrödinger’s seminal 1935 paper, and has to do with the apparent fuzzy nature of the reality described by the formalism through the wavefunction $$\psi$$ ψ. This issue, which I will be calling the Determinacy Problem, is distinct from the standard measurement problem of quantum mechanics, despite Schrödinger himself ends up conflating the two. I will argue that the (...) class='Hi'>Determinacy Problem is an exquisitely philosophical problem, for as it is standard when facing any phenomenon which appears to have indeterminate or fuzzy characteristics, the solutions available are to either blame the deficiencies of our language, or our lack of knowledge, or to blame the world itself. These three attitudes can already be found in the literature on quantum mechanics, either explicitly or implicitly, and they appear to motivate three very distinct research programs: high-dimensional realism, primitive ontology, and quantum indeterminacy. (shrink)

We characterize in terms of determinacy, the existence of the least inner model of “every real has a sharp”. We let #1 be the sharp function on the reals and define two classes of sets, * and *+, which lie strictly between β<ω2- and Δ. We show that the determinacy of * follows from L[#1] “every reak has a sharp”; and we show that the existence of indiscernibles for L[#1] is equivalent to a slightly stronger determinacy hypothesis, (...) the determinacy of *+. (shrink)

We show, assuming weak large cardinals, that in the context of games of length \ with moves coming from a proper class, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of L that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in (...) third, seventh, or \\)th order arithmetic. We also prove bounds on the strength of Borel determinacy for proper class games. These results answer questions of Gitman and Hamkins. (shrink)

We introduce the notion of -determinacy for Γ a pointclass and E an equivalence relation on a Polish space X. A case of particular interest is the case when E = EG is the shift-action o...

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss (...) a particularly popular approach to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A. $\mathrm{D}\mathrm{e}\mathrm{t}\text{\hspace{0.17em}}({\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D})$ ⇒ there exists an inner model with a strong cardinal. Theorem B. Det(AQI) ⇒ there exist type-1 mice and hence inner models with proper classes of strong cardinals. where ${\mathrm{\Pi }}_{1}^{1}-\mathrm{I}\mathrm{N}\mathrm{D}\phantom{\rule{0ex}{0ex}}$ (AQI) is the pointclass of boldface ${\mathrm{\Pi }}_{1}^{1}$ -inductive (respectively arithmetically quasi-inductive) sets of reals.

This paper distinguishes between definiteness and determinacy. Definiteness is seen as a morphological category which, in English, marks a uniqueness presupposition, while determinacy consists in denoting an individual. Definite descriptions are argued to be fundamentally predicative, presupposing uniqueness but not existence, and to acquire existential import through general type-shifting operations that apply not only to definites, but also indefinites and possessives. Through these shifts, argumental definite descriptions may become either determinate or indeterminate. The latter option is observed in (...) examples like ‘Anna didn’t give the only invited talk at the conference’, which, on its indeterminate reading, implies that there is nothing in the extension of ‘only invited talk at the conference’. The paper also offers a resolution of the issue of whether possessives are inherently indefinite or definite, suggesting that, like indefinites, they do not mark definiteness lexically, but like definites, they typically yield determinate readings due to a general preference for the shifting operation that produces them. (shrink)

We show that the semi-linear ordering principle for continuous functions implies the determinacy of all Wadge and Lipschitz games.

The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy “one level up” into the meta-theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This paper (...) argues that the challenge can be met. We show how the quantifier “there are infinitely many” can be uniquely determined in a naturalistically acceptable fashion and thus be used in the formulation of a theory of arithmetic. We compare the approach pursued here with Field’s justification of the same device and the popular strategy of invoking a second-order formalism, and argue that it is more robust than either of the alternative proposals. (shrink)

We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. (...) A consequence is the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy. (shrink)

We investigate a simple game paradigm for intuitionistic logic, inspired by Wajsberg’s implicit inhabitation algorithm and Beth tableaux. The principal idea is that one player, ∃ros, is trying to construct a proof in normal form while his opponent, ∀phrodite, attempts to build a counter-model. The determinacy of the game implies therefore both completeness and semantic cut-elimination.

From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of ℵ1 is constructible from a real.

We show under that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under if we additionally assume that the set of Borel codes for I‐positive sets is. If we do not assume, the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness similar to (...) the one introduced by Bagaria and Bosch [2], we show under without using that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is strongly proper assuming every set of reals is ∞‐Borel and there is no ω1‐sequence of distinct reals. In particular, the same conclusion holds in a Solovay model. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Belief and Context Determinacy in Interpreting FictionChristine Richards (bio)1Context Determinacy and the Interpretation of FictionThe Pragmatics of ReadingThe basic pragmatic structure of the reading of fiction has been described as a communicative context which has a speaker who performs the speech acts represented by the text and a hearer (addressee) to whom the speech acts are directed [Adams 12]. This model is based on the assumption that (...) the reader and the writer share not only a common language but a common context, which clearly presents problems for readers of fiction, where generic conventions as well as other cultural conventions of writing, such as satire and irony, or the separation of writers and readers by time, have to be contended with. The code model, which assumes that the writer and the reader share a common context, can be represented as shown in figure 1. Click for larger view View full resolutionFigure 1.W = writer, S = speaker, H = hearer, R = reader.However, it has also been argued [Eco, Role of the Reader 40; Eco, Semiotics 69; Richards 262] that no theory of a text is possible without a theory of contexts, since texts are processed and given meaning by a synthesis of propositions in them with those which originate in readers’ supplied contexts; and since the contexts of readers consist, first and foremost, of their background knowledge, beliefs, and culture, which are fundamentally interrelated, interpretation is inevitably determined by extralinguistic as well as linguistic factors. The context model might be represented as shown in figure 2. Click for larger view View full resolutionFigure 2.W = writer, N = narrator, R = reader.[End Page 81]The Extralinguistic ContextA good example of the role played by extralinguistic contexts in the interpretation of a text is to be found in the multiple interpretations of the sentence “The red sun was pasted in the sky like a wafer,” from Stephen Crane’s The Red Badge of Courage [64]. Susan Horton, in her discussion of accounts of this sentence [10–14], shows both how R. W. Stallman interprets Crane’s antiwar fiction as a Christian allegory where the word “wafer” functions as a symbol of the Eucharist [130], and how Marston La France rejects Stallman’s interpretation on the basis that Crane is not a Christian, and certainly not a Catholic [99–100]. La France’s rejection of Stallman’s Christian context for atheism is rooted in his interpretation of the sun as a repeated image in the co-text, which he believes signifies nature’s indifference to death, war and heroism. Here the lexemes “pasted” and “wafer” become images of a seal at the end of a legal document, implying the finality of death in war. Milne Holton interprets the image of the sun as a characteristic of the symbolic imagination of the character Henry Fleming rather than of the writer (Crane) [90–91].Horton documents but does not theorize the determining features of these contexts, but this problem can be tackled by applying Marcello Pagnini’s theory of “the introjection of the referent” [9] to explain how the contextual semes of readers, determined by different cultural systems, are introjected into texts. Following Jurij M. Lotman’s typology of cultural systems [Lotman and Uspenskij], Pagnini cites as relevant systems ethics, ideology, history, philosophy, myth, the law, anthropology, politics, ethnicity, literature, aesthetics, psychology, rhetoric and symbolism [34], which represent fundamentally interrelated complexes of systems that make up societies. If Pagnini’s model is taken with that of Richards [267], which conceives of interpretation as a synthesis of propositional inferences in texts with those drawn from readers’ contexts, these two models provide a methodological basis for explaining how the contexts of Stallman’s and La France’s conflicting interpretations derive from philosophically opposed belief systems within culture—Christianity and atheism. These models also help to explain Holton’s interpretation as one located within a different philosophical context, that of the convention of character analysis in fiction, which during the last few decades has developed into the more theoretically sophisticated field of narratology. It should be stressed, however, that interpretations themselves always take place within historical and interactive frameworks, which are influenced... (shrink)

We consider a seemingly weaker form of $\Delta ^{1}_{1}$ Turing determinacy.Let $2 \leqslant \rho < \omega _{1}^{\mathsf {CK}}$, $\textrm{Weak-Turing-Det}_{\rho }$ is the statement:Every $\Delta ^{1}_{1}$ set of reals cofinal in the Turing degrees contains two Turing distinct, $\Delta ^{0}_{\rho }$ -equivalent reals.We show in $\mathsf {ZF}^-$ : $\textrm{Weak-Turing-Det}_{\rho }$ implies: for every $\nu < \omega _{1}^{\mathsf {CK}}$ there is a transitive model ${M \models \mathsf {ZF}^{-} + \textrm{``}\aleph _{\nu } \textrm{ exists''.}}$ As a corollary:If every cofinal $\Delta ^{1}_{1}$ set (...) of Turing degrees contains both a degree and its jump, then for every $\nu < \omega_1^{\mathsf{CK}}$, there is atransitive model: $M \models \mathsf{ZF}^{-} + \textrm{``}\aleph_\nu \textrm{ exists''.}$ •With a simple proof, this improves upon a well-known result of Harvey Friedman on the strength of Borel determinacy.•Invoking Tony Martin’s proof of Borel determinacy, $\textrm{Weak-Turing-Det}_{\rho }$ implies $\Delta ^{1}_{1}$ determinacy.•We show further that, assuming $\Delta ^{1}_{1}$ Turing determinacy, or Borel Turing determinacy, as needed:–Every cofinal $\Sigma ^{1}_{1}$ set of Turing degrees contains a “hyp-Turing cone”: ${\{x \in \mathcal {D} \mid d_{0} \leqslant _{T} x \leqslant _{h} d_{0} \}}$.–For a sequence $_{k < \omega }$ of analytic sets of Turing degrees, cofinal in $\mathcal {D}$, $\bigcap _{k} A_{k}$ is cofinal in $\mathcal {D}$. (shrink)