It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theory. In return, the existence of such large sets turned out critical to settling open conjectures about (...) point sets.How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?In this essay I dissect one group of such two-way results. Their linchpin is the notion of measure.§1. Vitali's impossibility result. Our starting point is a problem in measure theory regarding the notion of “Lebesgue measure.” Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets. (shrink)

Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an L ω 1ω -interpretability theory. We also give a survey of this theory.

We characterize, in terms of determinacy, the existence of the least inner model of "every object of type k has a sharp." For k ∈ ω, we define two classes of sets, (Π 0 k ) * and (Π 0 k ) * + , which lie strictly between $\bigcup_{\beta and Δ(ω 2 -Π 1 1 ). Let ♯ k be the (partial) sharp function on objects of type k. We show that the determinancy of (Π 0 k ) * (...) follows from $L \lbrack\ sharp_k \rbrack \models "\text{every object of type} k \text{has a sharp},$ and we show that the existence of indiscernibles for L[ ♯ k ] is equivalent to a slightly stronger determinacy hypothesis, the determinacy of (Π 0 k ) * +. (shrink)

We consider fixed point logics, i.e., extensions of first order predicate logic with operators defining fixed points. A number of such operators, generalizing inductive definitions, have been studied in the context of finite model theory, including nondeterministic and alternating operators. We review results established in finite model theory, and also consider the expressive power of the resulting logics on infinite structures. In particular, we establish the relationship between inflationary and nondeterministic fixed point logics and second order logic, and we consider (...) questions related to the determinacy of games associated with alternating fixed points. (shrink)

Before one can construct scales of minimal complexity in the Real Core Model, K(R), one needs to develop the fine-structure theory of K(R). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice (...) satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(R). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(R) and the strength of the theory ZF + AD + ¬ DC R. (shrink)

We prove that, for any natural number n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}, we can find a finite alphabet Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} and a finitary language L over Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} accepted by a one-counter automaton, such that the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-power L∞:={w0w1…∈Σω∣∀i∈ωwi∈L}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (...) L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$\end{document}is Πn0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Pi }}^0_n$$\end{document}-complete. We prove a similar result for the class Σn0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}^0_n$$\end{document}. (shrink)

We prove a number of results on the determinacy of σ-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between σ-projective determinacy and the determinacy of certain classes of games of variable length <ω^2 (Theorem 2.4). We then give an elementary proof of the determinacy of σ-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof (...) to obtain proofs of the determinacy of σ-projective games of a given countable length and of games with payoff in the smallest σ-algebra containing the projective sets, from corresponding assumptions (Theorem 5.1, Theorem 5.4). (shrink)

Let ξ ≥ 1 be a countable ordinal. We study the Borel subsets of the plane that can be made [Formula: see text] by refining the Polish topology on the real line. These sets are called potentially [Formula: see text]. We give a Hurewicz-like test to recognize potentially [Formula: see text] sets.

The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...) combines Husserl's insights into the nature of perception and the phenomena of evidence and justification with case studies of the role of intuition in axiomatic set theory. (shrink)

Local sentences and the formal languages they define were introduced by Ressayre in. We prove that locally finite ω‐languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω‐languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non‐null recursive ordinal there exist some ‐complete and some ‐complete locally finite ω‐languages, and the supremum of the set of Borel ranks (...) of locally finite ω‐languages is the ordinal, which is strictly greater than the first non‐recursive ordinal. This gives an answer to the question of the topological complexity of locally finite ω‐languages, which was asked by Simonnet and also by Duparc, Finkel, and Ressayre in. Moreover we show that the topological complexity of a locally finite ω‐language defined by a local sentence φ may depend on the models of the Zermelo‐Fraenkel axiomatic system. Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω‐languages are ‐complete, hence highly undecidable. (shrink)

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

We propose a simple notion of "extender" for coding large elementary embeddings of models of set theory. As an application we present a self-contained proof of the theorem by D. Martin and J. Steel that infinitely many Woodin cardinals imply the determinacy of every projective set.

This paper is an attempt to convince anti-realists that their correct intuitions against the metaphysical inflationism derived from some versions of mathematical realism do not force them to embrace non-standard, epistemic approaches to truth and existence. It is also an attempt to convince mathematical realists that they do not need to implement their perfectly sound and judicious intuitions with the anti-intuitive developments that render full-blown mathematical realism into a view which even Gödel considered objectionable. I will argue for the following (...) two theses: that realism, in its standard characterization, is our default position, a position in agreement with our pre-theoretical intuitions and with the results of our best semantic theories, and that most of the metaphysical qualms usually related to it depends on a poor understanding of truth and existence as higher-order concepts. (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.

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by ∝(f) the separation rank of a real-valued functionfinB1(X), withXa Polish space. Notice that in [3], (...) Bourgain has provided a positive answer to this problem in the case ofKsatisfyingwithXa compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a functionf, then the possible discontinuities of the limit function reflect a local ℓ1-structure to the sequence (fn)n. More precisely the complexity of this ℓ1-structure increases as the complexity of the discontinuities offdoes. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between thec0-structure of a sequence (fn)nof uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal'sc0-theorem [11] and thec0-index theorem [2] are consequences of this interaction.Passing to the case where either (fn)nare not continuous orXis a non-compact Polish space, this nice interaction is completely lost. (shrink)

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅' of the probability that the output be in some set.

A standard result about hyperdegrees is proved (under PD) for all ▵ 1 2n + 1 -degrees.

We discuss in the paper the following problem: Given a function in a given Baire class, into "how many" (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.

This paper introduces an extension A of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(α) on Baire space with the property that every constructive partial functional defined on {α : R(α)} is continuous there. The domains of continuity for A coincide with the stable relations (those equivalent in A to their double negations), while every relation R(α) (...) is equivalent in A to ∃αA(α, β) for some stable A(α, β) (which belongs to the classical analytical hierarchy). The logic of A is intuitionistic. The axioms of A include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems, A is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that A preserves the constructive sense of "or" and "there exists.". (shrink)

We develop a variant of Least Fixed Point logic based on First Orderlogic with a relaxed version of guarded quantification. We develop aGame Theoretic Semantics of this logic, and find that under reasonableconditions, guarding quantification does not reduce the expressibilityof Least Fixed Point logic. But we also find that the guarded version ofa least fixed point algorithm may have a greater time complexity thanthe unguarded version, by a linear factor.

Maddy's notion of restrictiveness has many problematic aspects, one of them being that it is almost impossible to show that a theory is not restrictive. In this note the author addresses a crucial question of Martin Goldstern (Vienna) and points to some directions of future research.

The present article, which is a revised version of part of [Hu1], deals with various relations between models which might serve as exact formulations for the vague concept "similar" or "almost isomorphic". One natural class of such formulations is equivalence in a given logic. Another way to express similarity is by potential isomorphism, i.e., isomorphism in some extension of the set-theoretic universe. The class of extensions may be restricted to give different notions of potential isomorphism. A third method is to (...) study the winning strategies for an Ehrenfeucht-Fraisse-game played between the two models, and the properties of the resulting equivalence and nonequivalence trees. The basic question studied here is whether one such notion of similarity implies another. Some implications and counterexamples listed in this part are previously known or trivial, but all are mentioned for completeness' sake. Only models of cardinality ℵ 1 are considered. Some results are therefore connected with the Continuum Hypothesis. (shrink)

First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete (...) axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized. (shrink)

I examine various claims to the effect that Cantor's Continuum Hypothesis and other problems of higher set theory are ill-posed questions. The analysis takes into account the viability of the underlying philosophical views and recent mathematical developments.

Convinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

The choice construct (choose x: φ(x)) is useful in software specifications. We study extensions of first-order logic with the choice construct. We prove some results about Hilbert's ε operator, but in the main part of the paper we consider the case when all choices are independent.

This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space $${\mathcal {X}}$$ X, and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space $$\mathbb {N}\times {\mathcal {X}}$$ N × X.

We prove that, for any natural number \, we can find a finite alphabet \ and a finitary language L over \ accepted by a one-counter automaton, such that the \-power $$\begin{aligned} L^\infty :=\{ w_0w_1\ldots \in \Sigma ^\omega \mid \forall i\in \omega ~~w_i\in L\} \end{aligned}$$is \-complete. We prove a similar result for the class \.

Developing an analogue of Solovay reducibility in the higher recursion setting, we show that results from the classical computably enumerable case can be extended to the new context.

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δta 2 is just large enough to include several types of pointsets in Euclidean spaces ℝ k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class ΔB 2 and Ershov's hierarchy in the class Δ0 2 of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δta 2. This is based on suitable (...) characterizations of the sets from Δta 2 which are obtained in a close analogy to those of the ΔB 2 sets as well as those of the Δ0 2 sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω 2. Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented. (shrink)

We carry out a systematic study of decidability for theories of real vector spaces, inner product spaces, and Hilbert spaces and of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list turn out to be decidable while the theories for list are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and (...) purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable. (shrink)

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L.The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and (...) descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL ⇒ HODL ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals. (shrink)

First we prove that the set of countable linear orders of the form I + I form a complete analytic set. As a consequence of this we improve a result of Humke and Laczkovich, who showed in [HL] that the set of functions of the form f ⚬ f form a true analytic set in C[0, 1]. We show that these functions form a complete analytic set, solving a problem mentioned on p. 215 of [K1] and on p. 4 of (...) [B]. (shrink)

We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals thatif the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter are determined;all games of length $\omega_1$ with payoff constructible relative to the play are determined; andif the Continuum Hypothesis holds, then there is a model of (...) ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined. (shrink)