Results for 'regular reals'
Regular Reals.Guohua Wu - 2005 - Mathematical Logic Quarterly 51 (2):111-119.details
Say that α is an n-strongly c. e. real if α is a sum of n many strongly c. e. reals, and that α is regular if α is n-strongly c. e. for some n. Let Sn be the set of all n-strongly c. e. reals, Reg be the set of regular reals and CE be the set of c. e. reals. Then we have: S1 ⊂ S2 ⊂ · · · ⊂ Sn ⊂ (...) · · · ⊂ ⊂ Reg ⊂ CE. This gives a hierarchy of the c. e. reals. We also study the regularity of the d. c. e. reals. (shrink)
Areas of Mathematics in Philosophy of Mathematics
Regular Subalgebras of Complete Boolean Algebras.Aleksander Błaszczyk & Saharon Shelah - 2001 - Journal of Symbolic Logic 66 (2):792-800.details
It is proved that the following conditions are equivalent: (a) there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, (b) there exists a nowhere dense ultrafilter on ω. Therefore, the existence of such algebras is undecidable in ZFC. In "forcing language" condition (a) says that there exists a non-trivial σ-centered forcing not adding Cohen reals.
Σ12-Sets of Reals.Jaime I. Ihoda - 1988 - Journal of Symbolic Logic 53 (2):636 - 642.details
We prove that the only implications between four notions for Σ 1 2 -sets of reals are $\Sigma^1_2-\text{measurability} \Rightarrow \Sigma^1_2-\text{categoricity} \big\downarrow \Sigma^1_2-\text{Ramsey} \Rightarrow \Sigma^1_2-K_\sigma-\text{regular}$.
Model Theory in Logic and Philosophy of Logic
Exact Equiconsistency Results for Δ 3 1 -Sets of Reals.Haim Judah - 1992 - Archive for Mathematical Logic 32 (2):101-112.details
We improve a theorem of Raisonnier by showing that Cons(ZFC+every Σ 2 1 -set of reals in Lebesgue measurable+every Π 2 1 -set of reals isK σ-regular) implies Cons(ZFC+there exists an inaccessible cardinal). We construct, fromL, a model where every Δ 3 1 -sets of reals is Lebesgue measurable, has the property of Baire, and every Σ 2 1 -set of reals isK σ-regular. We prove that if there exists a Σ n+1 1 unbounded (...) filter on ω, then there exists a nonK σ-regular Π 2 1 -subset. (shrink)
Changing Cardinal Invariants of the Reals Without Changing Cardinals or the Reals.Heike Mildenberger - 1998 - Journal of Symbolic Logic 63 (2):593-599.details
We show: The procedure mentioned in the title is often impossible. It requires at least an inner model with a measurable cardinal. The consistency strength of changing b and d from a regular κ to some regular δ < κ is a measurable of Mitchell order δ. There is an application to Cichon's diagram.
Axioms of Set Theory in Philosophy of Mathematics
Cardinals and Ordinals in Philosophy of Mathematics
$Sigma^1_2$-Sets of Reals.Jaime I. Ihoda - 1988 - Journal of Symbolic Logic 53 (2):636-642.details
We prove that the only implications between four notions for $\Sigma^1_2$-sets of reals are $\Sigma^1_2-\text{measurability} \Rightarrow \Sigma^1_2-\text{categoricity} \big\downarrow \Sigma^1_2-\text{Ramsey} \Rightarrow \Sigma^1_2-K_\sigma-\text{regular}$.
Models with Second Order Properties IV. A General Method and Eliminating Diamonds.Saharon Shelah - 1983 - Annals of Pure and Applied Logic 25 (2):183-212.details
We show how to build various models of first-order theories, which also have properties like: tree with only definable branches, atomic Boolean algebras or ordered fields with only definable automorphisms. For this we use a set-theoretic assertion, which may be interesting by itself on the existence of quite generic subsets of suitable partial orders of power λ + , which follows from ♦ λ and even weaker hypotheses . For a related assertion, which is equivalent to the morass see Shelah (...) and Stanley [16]. The various specific constructions serve also as examples of how to use this set-theoretic lemma. We apply the method to construct rigid ordered fields, rigid atomic Boolean algebras, trees with only definable branches; all in successors of regular cardinals under appropriate set- theoretic assumptions. So we are able to answer the following algebraic question. Saltzman's Question . Is there a rigid real closed field, which is not a subfield of the reals? (shrink)
Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic
Martin's Maximum and Definability in H.Paul B. Larson - 2008 - Annals of Pure and Applied Logic 156 (1):110-122.details
In [P. Larson, Martin's Maximum and the axiom , Ann. Pure App. Logic 106 135–149], we modified a coding device from [W.H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter & Co, Berlin, 1999] and the consistency proof of Martin's Maximum from [M. Foreman, M. Magidor, S. Shelah, Martin's Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annal. Math. 127 1–47] to show that from a supercompact limit of supercompact cardinals one could force (...) Martin's Maximum to hold while the axiom fails. Here we modify that argument to prove a stronger fact, that Martin's Maximum is consistent with the existence of a wellordering of the reals definable in H without parameters, from the same large cardinal hypothesis. In doing so we give a much simpler proof of the original result. (shrink)
A diamond-plus principle consistent with AD.Daniel W. Cunningham - 2020 - Archive for Mathematical Logic 59 (5-6):755-775.details
After showing that \ refutes \ for all regular cardinals \, we present a diamond-plus principle \ concerning all subsets of \. Using a forcing argument, we prove that \ holds in Steel's core model \}}\), an inner model in which the axiom of determinacy can hold. The combinatorial principle \ is then extended, in \}}\), to successor cardinals \ and to certain cardinals \ that are not ineffable. Here \ is the supremum of the ordinals that are the (...) surjective image of the set of reals \. (shrink)
Inner Models From Extended Logics: Part 1.Juliette Kennedy, Menachem Magidor & Jouko Väänänen - forthcoming - Journal of Mathematical Logic:2150012.details
If we replace first-order logic by second-order logic in the original definition of Gödel's inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or (...) stationary logic [6]. Our first set of results show that both [Formula: see text] and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between [Formula: see text] and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals [Formula: see text] of [Formula: see text] are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals, the Continuum Hypothesis is true in the relativized model. (shrink)
Noetherian Varieties in Definably Complete Structures.Tamara Servi - 2008 - Logic and Analysis 1 (3-4):187-204.details
We prove that the zero-set of a C ∞ function belonging to a noetherian differential ring M can be written as a finite union of C ∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C ∞ functions over the reals, but more generally for (...) definable C ∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes o-minimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C ∞ functions over the reals, containing non-analytic functions. (shrink)
Logics in Logic and Philosophy of Logic
Nonclassical Logics in Logic and Philosophy of Logic
A Cardinal Preserving Extension Making the Set of Points of Countable V Cofinality Nonstationary.Moti Gitik, Itay Neeman & Dima Sinapova - 2007 - Archive for Mathematical Logic 46 (5-6):451-456.details
Assuming large cardinals we produce a forcing extension of V which preserves cardinals, does not add reals, and makes the set of points of countable V cofinality in κ+ nonstationary. Continuing to force further, we obtain an extension in which the set of points of countable V cofinality in ν is nonstationary for every regular ν ≥ κ+. Finally we show that our large cardinal assumption is optimal.
Reals by Abstraction.Bob Hale - 2000 - Philosophia Mathematica 8 (2):100--123.details
On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...) explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities. (shrink)
Analysis in Philosophy of Mathematics
Frege: Abstraction Principles in 20th Century Philosophy
Frege: Real Numbers in 20th Century Philosophy
Logicism in Mathematics in Philosophy of Mathematics
Mathematical Neo-Fregeanism in Philosophy of Mathematics
Quantities in Metaphysics
Direct download (11 more)
Reals by Abstraction.Bob Hale - 2000 - The Proceedings of the Twentieth World Congress of Philosophy 6:197-207.details
While Frege's own attempt to provide a purely logical foundation for arithmetic failed, Hume's principle suffices as a foundation for elementary arithmetic. It is known that the resulting system is consistent—or at least if second-order arithmetic is. Some philosophers deny that HP can be regarded as either a truth of logic or as analytic in any reasonable sense. Others—like Crispin Wright and I—take the opposed view. Rather than defend our claim that HP is a conceptual truth about numbers, I explain (...) one way it may be possible to extend our view beyond elementary arithmetic to encompass the theory of real numbers. My approach has affinities to the leading ideas of Frege's own treatment of the reals, although differing in one fundamental way. I attempt, like the HP approach to elementary arithmetic, to obtain the reals very directly by means of abstraction principles without any essential reliance on a theory of sets. This is the most natural way of extending the neo-Fregean position to the reals. (shrink)
Random Reals, the Rainbow Ramsey Theorem, and Arithmetic Conservation.Chris J. Conidis & Theodore A. Slaman - 2013 - Journal of Symbolic Logic 78 (1):195-206.details
We investigate the question "To what extent can random reals be used as a tool to establish number theoretic facts?" Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem (...) is not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$. (shrink)
New Reals: Can Live with Them, Can Live Without Them.Martin Goldstern & Jakob Kellner - 2006 - Mathematical Logic Quarterly 52 (2):115-124.details
We give a self-contained proof of the preservation theorem for proper countable support iterations known as "tools-preservation", "Case A" or "first preservation theorem" in the literature. We do not assume that the forcings add reals.
On the Constructive Dedekind Reals.Robert S. Lubarsky & Michael Rathjen - 2008 - Logic and Analysis 1 (2):131-152.details
In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which (...) the Cauchy reals form a set while the Dedekind reals constitute a proper class. (shrink)
Intuitionistic Logic in Logic and Philosophy of Logic
Random Reals and Possibly Infinite Computations Part I: Randomness in ∅'.Verónica Becher & Serge Grigorieff - 2005 - Journal of Symbolic Logic 70 (3):891-913.details
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.
Δ31 Reals.René David - 1982 - Annals of Mathematical Logic 23 (2-3):121-125.details
Partition Reals and the Consistency of T > Add(R).Kyriakos Keremedis - 1993 - Mathematical Logic Quarterly 39 (1):545-550.details
We show that it is consistent with ZFC that the additivity number add of the ideal of meager sets of the real line is strictly greater than the tower number t of the reals. MSC: 03E35, 54D20.
Symmetry Arguments Against Regular Probability: A Reply to Recent Objections.Matthew Parker - 2018 - European Journal for Philosophy of Science 9 (1):8.details
A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson's (2007) "isomorphic" events are not in fact isomorphic, but Howson (...) is speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson's physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)
Infinitesimals and Probability in Philosophy of Probability
Probabilistic Principles, Misc in Philosophy of Probability
Amoeba Reals.Haim Judah & Miroslav Repickẏ - 1995 - Journal of Symbolic Logic 60 (4):1168-1185.details
We define the ideal with the property that a real omits all Borel sets in the ideal which are coded in a transitive model if and only if it is an amoeba real over this model. We investigate some other properties of this ideal. Strolling through the "amoeba forest" we gain as an application a modification of the proof of the inequality between the additivities of Lebesgue measure and Baire category.
Countable OD Sets of Reals Belong to the Ground Model.Vladimir Kanovei & Vassily Lyubetsky - 2018 - Archive for Mathematical Logic 57 (3-4):285-298.details
It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.
Against Regular and Irregular Characterizations of Mechanisms.Lane DesAutels - 2011 - Philosophy of Science 78 (5):914-925.details
This article addresses the question of whether we should conceive of mechanisms as productive of change in a regular way. I argue that, if mechanisms are characterized as fully regular, on the one hand, then not enough processes will count as mechanisms for them to be interesting or useful. If no appeal to regularity is made at all in their characterization, on the other hand, then mechanisms can no longer be useful for grounding prediction and supporting intervention strategies. (...) I conclude that, if the New Mechanistic Philosophy is to be successful, a stochastic characterization of mechanisms must be adopted. (shrink)
Explanation in Biology in Philosophy of Biology
Mechanistic Explanation in General Philosophy of Science
Needed Reals and Recursion in Generic Reals.Andreas Blass - 2001 - Annals of Pure and Applied Logic 109 (1-2):77-88.details
We consider sets of reals that are "adequate" in various senses, for example dominating or unbounded or splitting or non-meager. Call a real x "needed" if every adequate set contains a real in which x is recursive. We characterize the needed reals for numerous senses of "adequate." We also consider, for various notions of forcing that add reals, the problem of characterizing the ground-model reals that are recursive in generic reals.
Complexity of Reals in Inner Models of Set Theory.Boban Velickovic & W. Hugh Woodin - 1998 - Annals of Pure and Applied Logic 92 (3):283-295.details
We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals (...) are an uncountable Fσ set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly 1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry. (shrink)
Hechler Reals.Grzegorz Łabędzki & Miroslav Repický - 1995 - Journal of Symbolic Logic 60 (2):444-458.details
We define a σ-ideal J D on the set of functions ω ω with the property that a real x ∈ ω ω is a Hechler real over V if and only if x omits all Borel sets in J D . In fact we define a topology D on ω ω related to Hechler forcing such that J D is the family of first category sets in D. We study cardinal invariants of the ideal J D.
Polish Philosophy in European Philosophy
Regular Opens in Constructive Topology and a Representation Theorem for Overlap Algebras.Francesco Ciraulo - 2013 - Annals of Pure and Applied Logic 164 (4):421-436.details
Giovanni Sambin has recently introduced the notion of an overlap algebra in order to give a constructive counterpart to a complete Boolean algebra. We propose a new notion of regular open subset within the framework of intuitionistic, predicative topology and we use it to give a representation theorem for overlap algebras. In particular we show that there exists a duality between the category of set-based overlap algebras and a particular category of topologies in which all open subsets are (...) class='Hi'>regular. (shrink)
Δ12-Sets of Reals.Jaime I. Ihoda & Saharon Shelah - 1989 - Annals of Pure and Applied Logic 42 (3):207-223.details
On Regular Groups and Fields.Tomasz Gogacz & Krzysztof Krupiński - 2014 - Journal of Symbolic Logic 79 (3):826-844.details
Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. LetKbe a regular field which is not generically stable and letpbe its global generic type. We observe that ifKhas a finite extensionLof degreen, thenPhas unbounded orbit (...) under the action of the multiplicative group ofL.Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable. (shrink)
Schnorr Trivial Reals: A Construction. [REVIEW]Johanna N. Y. Franklin - 2008 - Archive for Mathematical Logic 46 (7-8):665-678.details
A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}$ (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if ${{\bf h'} \geq_T {\bf 0''}}$ (...) , then h contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. (shrink)
Regular Updating.Alain Chateauneuf, Thibault Gajdos & Jean-Yves Jaffray - 2011 - Theory and Decision 71 (1):111-128.details
We study the Full Bayesian Updating rule for convex capacities. Following a route suggested by Jaffray (IEEE Transactions on Systems, Man and Cybernetics 22(5):1144–1152, 1992), we define some properties one may want to impose on the updating process, and identify the classes of (convex and strictly positive) capacities that satisfy these properties for the Full Bayesian Updating rule. This allows us to characterize two parametric families of convex capacities: ${(\varepsilon,\delta)}$ -contaminations (which were introduced, in a slightly different form, by Huber (...) (Robust Statistics, Wiley, New York, 1981)) and ${\varepsilon}$ -contaminations. (shrink)
Bayesian Reasoning, Misc in Philosophy of Probability
Updating Principles in Philosophy of Probability
Uncountable Trees and Cohen -Reals.Giorgio Laguzzi - 2019 - Journal of Symbolic Logic 84 (3):877-894.details
We investigate some versions of amoeba for tree-forcings in the generalized Cantor and Baire spaces. This answers [10, Question 3.20] and generalizes a line of research that in the standard case has been studied in [11], [13], and [7]. Moreover, we also answer questions posed in [3] by Friedman, Khomskii, and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show ${\bf{\Sigma }}_1^1$-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong (...) relationship between the Ramsey and the Baire properties, in slight contrast with the standard case. (shrink)
A Transfinite Hierarchy of Reals.George Barmpalias - 2003 - Mathematical Logic Quarterly 49 (2):163-172.details
We extend the hierarchy defined in [5] to cover all hyperarithmetical reals. An intuitive idea is used or the definition, but a characterization of the related classes is obtained. A hierarchy theorem and two fixed point theorems are presented.
Recursion Theory on the Reals and Continuous-Time Computation.Christopher Moore - 1996 - Theoretical Computer Science 162:23--44.details
Computability in Philosophy of Computing and Information
Mapping a Set of Reals Onto the Reals.Arnold W. Miller - 1983 - Journal of Symbolic Logic 48 (3):575-584.details
In this paper we show that it is consistent with ZFC that for any set of reals of cardinality the continuum, there is a continuous map from that set onto the closed unit interval. In fact, this holds in the iterated perfect set model. We also show that in this model every set of reals which is always of first category has cardinality less than or equal to ω 1.
Approximation Representations for Reals and Their Wtt-Degrees.George Barmpalias - 2004 - Mathematical Logic Quarterly 50 (45):370-380.details
We study the approximation properties of computably enumerable reals. We deal with a natural notion of approximation representation and study their wtt-degrees. Also, we show that a single representation may correspond to a quite diverse variety of reals.
Deciding Regular Grammar Logics with Converse Through First-Order Logic.Stéphane Demri & Hans De Nivelle - 2005 - Journal of Logic, Language and Information 14 (3):289-329.details
We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar (...) logics with converse. The class of regular grammar logics includes numerous logics from various application domains. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Other logics that can be translated into GF2 include nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed-point operators. (shrink)
Regular Universes and Formal Spaces.Erik Palmgren - 2006 - Annals of Pure and Applied Logic 137 (1-3):299-316.details
We present an alternative solution to the problem of inductive generation of covers in formal topology by using a restricted form of type universes. These universes are at the same time constructive analogues of regular cardinals and sets of infinitary formulae. The technique of regular universes is also used to construct canonical positivity predicates for inductively generated covers.
Fine-Tuning in Cosmology in Philosophy of Physical Science
Regular Relations for Temporal Propositions.T. Fernando - unknowndetails
Relations computed by finite-state transducers are applied to interpret temporal propositions in terms of strings representing finite contexts or situations. Carnap–Montague intensions mapping indices to extensions are reformulated as relations between strings that can serve as indices and extensions alike. Strings are related according to information content, temporal span and granularity, the bounds on which reflect the partiality of natural language statements. That partiality shapes not only strings-as-extensions (indicating what statements are about) but also strings-as-indices (underlying truth conditions).
Propositions, Misc in Philosophy of Language
Regular Probability Comparisons Imply the Banach–Tarski Paradox.Alexander R. Pruss - 2014 - Synthese 191 (15):3525-3540.details
Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: there can be nonmeasurable events that have no probability at all and on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice. We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient to (...) prove the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory. (shrink)
Symmetry Arguments Against Regular Probability: A Reply to Recent Objections.Matthew Parker - 2019 - European Journal for Philosophy of Science 9 (1):1-21.details
A probability distribution is regular if it does not assign probability zero to any possible event. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson and Benci et al. have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson's "isomorphic" events are not in fact isomorphic, but Howson (...) is speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson's physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)
Hyper-Regular Lattice-Ordered Groups.Daniel Gluschankof & François Lucas - 1993 - Journal of Symbolic Logic 58 (4):1342-1358.details
A Constructive Look at Generalised Cauchy Reals.Peter M. Schuster - 2000 - Mathematical Logic Quarterly 46 (1):125-134.details
We investigate how nonstandard reals can be established constructively as arbitrary infinite sequences of rationals, following the classical approach due to Schmieden and Laugwitz. In particular, a total standard part map into Richman's generalised Dedekind reals is constructed without countable choice.
$$AD_{Mathbb {R}}$$ A D R Implies That All Sets of Reals Are $$Theta $$ Θ Universally Baire.Grigor Sargsyan - 2021 - Archive for Mathematical Logic 60 (1-2):1-15.details
We show that assuming the determinacy of all games on reals, every set of reals is \ universally baire.
Strongly Dominating Sets of Reals.Michal Dečo & Miroslav Repický - 2013 - Archive for Mathematical Logic 52 (7-8):827-846.details
We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573–1581, 1995). We prove that for every ${\kappa < \mathfrak{b}}$ a ${\kappa}$ -Suslin set ${A\subseteq{}^\omega\omega}$ is strongly dominating if and only if A has a Laver perfect subset. We also investigate the structure of the class l of Baire sets for the Laver category base and compare the σ-ideal of sets which are not strongly dominating with the Laver ideal l (...) 0. (shrink)
Connexive Extensions of Regular Conditional Logic.Yale Weiss - 2019 - Logic and Logical Philosophy 28 (3):611-627.details
The object of this paper is to examine half and full connexive extensions of the basic regular conditional logic CR. Extensions of this system are of interest because it is among the strongest well-known systems of conditional logic that can be augmented with connexive theses without inconsistency resulting. These connexive extensions are characterized axiomatically and their relations to one another are examined proof-theoretically. Subsequently, algebraic semantics are given and soundness, completeness, and decidability are proved for each system. The semantics (...) is also used to establish independence results. Finally, a deontic interpretation of one of the systems is examined and defended. (shrink)
Unbounded and Dominating Reals in Hechler Extensions.Justin Palumbo - 2013 - Journal of Symbolic Logic 78 (1):275-289.details
We give results exploring the relationship between dominating and unbounded reals in Hechler extensions, as well as the relationships among the extensions themselves. We show that in the standard Hechler extension there is an unbounded real which is dominated by every dominating real, but that this fails to hold in the tree Hechler extension. We prove a representation theorem for dominating reals in the standard Hechler extension: every dominating real eventually dominates a sandwich composition of the Hechler real (...) with two ground model reals that monotonically converge to infinity. We apply our results to negatively settle a conjecture of Brendle and Löwe (Conjecture 15 of [4]). We also answer a question due to Laflamme. (shrink)
1\ Sets of Reals.J. Bagaria & W. H. Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.details
▵13-Sets of Reals.Haim Judah & Saharon Shelah - 1993 - Journal of Symbolic Logic 58 (1):72 - 80.details
We build models where all $\underset{\sim}{\triangle}^1_3$ -sets of reals are measurable and (or) have the property of Baire and (or) are Ramsey. We will show that there is no implication between any of these properties for $\underset{\sim}{\triangle}^1_3$ -sets of reals.
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