G is the result of adjoining the schema (qAA)qA to K; the axioms of G* are the theorems of G and the instances of the schema qAA and the sole rule of G* is modus ponens. A sentence is -provable if it is provable in P(eano) A(rithmetic) by one application of the -rule; equivalently, if its negation is -inconsistent in PA. Let -Bew(x) be the natural formalization of the notion of -provability. For any modal sentence A and function mapping sentence (...) letters to sentences of PA, inductively define A by: p = (p) (p a sentence letter); = ; (AB)su}= (A B); and (qA)= -Bew(A )(S) is the numeral for the Gödel number of the sentence S). Then, applying techniques of Solovay (Israel Journal of Mathematics 25, pp. 287–304), we prove that for every modal sentence A, G A iff for all , PA A ; and for every modal sentence A, G* A iff for all , A is true. (shrink)

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 \.

We study a generalization of the splitting number s to uncountable cardinals. We prove that $\mathfrak{s}(\kappa) > \kappa^+$ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s}(\aleph_\omega) > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes${\cal D}_2 \left( {{\bf{\Pi }}_3^0 } \right)$and${\cal D}_\omega \left( {{\bf{\Pi }}_3^0 } \right)$, that is, the class of sets which are 2-differences (respectively,ω-differences) of${\bf{\Pi }}_3^0 $sets.

In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too (...) large to be counted by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)

Let $\omega$ be the first infinite ordinal (or the set of all natural numbers) with the usual order $<$ . In § 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of $\omega$ , whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In § 2 we construct several $\lambda$ -Archimedean ultrapowers of $\omega$ under (...) some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a $\lambda$ -Archimedean ultrapower of $\omega$ for some uncountable cardinal $\lambda$ . This answers a question in [8], modulo the assumption of measurability. (shrink)

For ${b \in {^{\omega}}{\omega}}$ , let ${\mathfrak{c}^{\exists}_{b, 1}}$ be the minimal number of functions (or slaloms with width 1) to catch every functions below b in infinitely many positions. In this paper, by using the technique of forcing, we construct a generic model in which there are many coefficients ${\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}$ with pairwise different values. In particular, under the assumption that a weakly inaccessible cardinal exists, we can construct a generic model in which there are continuum many (...) coefficients ${\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}$ with pairwise different values. In conjunction with these results, we give a generic model in which there are many Yorioka’s ideals ${\mathcal{I}_{f_\alpha}}$ with pairwise different covering numbers. (shrink)

The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...) a \-cofinal set. The small uncountable cardinal \ is introduced. Consequences of \ and of \ are explored; in particular, both imply \. Here \ is the reaping number, and is also the least cardinality of a \-base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities and there exist simple \-points and \-points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on \ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold. The strongest of these, Axiom 1, is that for every free ultrafilter \ and for every \-unbounded \-chain C of increasing functions in \, C is also unbounded in the ultraproduct \. The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem. (shrink)

In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of Godel's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well-known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the ω-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory Z. (...) On the other hand, he claims that Z with its first-order logic is not strong enough to allow a formalization of such an ω-rule. This would explain why the unprovability of $\operatorname{Con}(Z)$ in Z does not imply that the consistency of Z cannot be proved by finitistic means. We show that Detlefsen's proposal is unacceptable as originally formulated in [1], but that a reasonable modification of the rule he suggest leads to a partial program already studied for many years. We investigate the scope of such a program in terms of proof-theoretic reducibilities. We also show that this partial program encompasses mathematically important theories studied in the "Reverse Mathematics" program. In order to investigate the provability with such a modified rule, we define new consistency and provability predicates which are weaker than the usual ones. We then investigate their properties, including a few that have no apparent philosophical significance but compare interestingly with the properties of the program based on the iteration of our ω-rule. We determine some of the limitations of such programs, pointing out that these limitations partly explain why partial programs that have been successfully carried out use quite different and substantially more radical extensions of finitistic methods with more general forms of restricted reasoning. (shrink)

We prove that for sufficiently large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $n$\end{document}, there are tautologies of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $O(n)$\end{document} that require proofs containing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Omega( n / \log n )$\end{document} lines in axiomatic systems of propositional logic based on the rules of substitution and detachment.

We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω 2-Rowbottom cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 measurable cardinals”. (...) We also discuss some generalizations of these results. (shrink)

Let T be a complete countable first-order theory such that every ultrapower of a model of T is saturated. If T has a model omitting a type p in every cardinality $ then T has a model omitting p in every cardinality. There is also a related theorem, and an example showing the $\beth_\omega$ cannot be improved.

Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If T satisfies $(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory (...) in L κ +, ω of cardinality κ at most, and φ (x, y) ∈ L λ +, ω }. This concept is interesting because of THEOREM 1. Let $T \subseteq L_{\kappa^+,\omega}$ of cardinality ≤ κ, and φ (x̄, ȳ) ∈ L λ +, ω . If $(\forall\mu then $(\forall_\chi > \kappa) I(\chi, T) = 2^\chi$ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ). Many years ago the second author proved that $\mu^\ast (\lambda, \kappa) \leq \beth_{(2^\lambda)^+}$ . Here we continue that work by proving. THEOREM 2. $\mu^\ast (\lambda, \aleph_0) = \beth_{\lambda^+}$ . THEOREM 3. For every κ ≤ λ we have $\mu^\ast (\lambda, \kappa) \leq \beth)_{(\lambda^\kappa)}^+$ . For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem. THEOREM 4. For every $\kappa \leq \lambda, T \subseteq L_{\kappa^+,\omega}$ , and any set of formulas $\Lambda \subseteq L_{\lambda^+,\omega}$ such that $\Lambda \subseteq L_{\kappa^+,\omega}$ , if T is (Λ,μ)-unstable for μ satisfying μ μ * (λ, κ) = μ then T is Λ-unstable (i.e. for every χ ≥ λ, T is (Λ, χ)-unstable). Moreover, T is L κ +, ω -unstable. In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers. (shrink)

Let σ(U) denote the number of automorphisms of a model U of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an U with $\omega_1 < \sigma(\mathfrak{U}) < 2^{\omega_1}$. We study the sufficiency of some conditions for σ(U) = 2ω1 . These conditions are analogous to conditions studied by D. Kueker in connection with countable models.

Let $\sigma$ denote the number of automorphisms of a model $\mathfrak{U}$ of power $\omega_1$. We derive a necessary and sufficient condition in terms of trees for the existence of an $\mathfrak{U}$ with $\omega_1 < \sigma < 2^{\omega_1}$. We study the sufficiency of some conditions for $\sigma = 2^{\omega_1}$. These conditions are analogous to conditions studied by D. Kueker in connection with countable models.

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it (...) may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations$\Omega ^{n}(\bar {\xi })$defined with the help of numeral exponents. Since$Num$(x) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with “insuperable difficulties.” This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that theN-operator$N(\bar {\xi })$and recursively defined (...) arithmetic operators$\Omega ^{n}(\bar {\xi })$are not different in kind, we can address Ramsey’s problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction. (shrink)

We show it is consistent with ZFC that the Hanf number of Ellentuck's Souslin logic should be exactly $\beth_{\omega_2}$.

We prove that every model of $T = \mathrm{Th}(\omega, countable) has an end extension; and that every countable theory with an infinite order and Skolem functions has 2 ℵ 0 nonisomorphic countable models; and that if every model of T has an end extension, then every |T|-universal model of T has an end extension definable with parameters.

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 analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set R of real numbers have the covering property of Menger if, and only if, each filter on ω associated with its (...) countable ω-cover is a P + filter. (shrink)

Let $\mathcal {I}$ be an ideal on $\omega $. For $f,\,g\in \omega ^{\omega }$ we write $f \leq _{\mathcal {I}} g$ if $f(n) \leq g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal {I}$. Moreover, we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ (in particular, $\mathcal {D}_{\mathrm {Fin}}$ denotes the family of all finite-to-one functions).We examine cardinal numbers $\mathfrak {b}(\geq _{\mathcal (...) {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ and $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq _{\mathcal {I}}$ sets in $\mathcal {D}_{\mathrm {Fin}}$ and $\mathcal {D}_{\mathcal {I}}$, respectively. For a maximal ideal $\mathcal {I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.We show that $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$ for all ideals $\mathcal {I}$ with the Baire property and that $\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$ for all coanalytic weak P-ideals (this class contains all $\bf {\Pi ^0_4}$ ideals). What is more, we give examples of Borel (even $\bf {\Sigma ^0_2}$ ) ideals $\mathcal {I}$ with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$ as well as with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$.We also study cardinals $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$ describing the smallest sizes of sets in $\mathcal {D}_{\mathcal {K}}$ not bounded from below with respect to the preorder $\leq _{\mathcal {I}}$ by any member of $\mathcal {D}_{\mathcal {J}}\!$. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN. (shrink)

In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides (...) with the classical tree-embeddability, but in this article we are interested in more general situations. We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ϑΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega}}$$\end{document}. Somewhat surprisingly, changing a little bit the well-partially ordered addresses, the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ϑΩΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega^\Omega}}$$\end{document}. Our results contribute to the research program on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings. (shrink)

This paper investigates properties of \(\sigma \) -closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter \({\mathcal {U}}\) for _n_-tuples, denoted \(t({\mathcal {U}},n)\), is the smallest number _t_ such that given any \(l\ge 2\) and coloring \(c:[\omega ]^n\rightarrow l\), there is a member \(X\in {\mathcal {U}}\) such that the restriction of _c_ to \([X]^n\) has no more than _t_ colors. Many well-known \(\sigma \) -closed forcings are known to generate ultrafilters with finite (...) Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by \(\sigma \) -closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings \({\mathcal {P}}(\omega ^k)/\mathrm {Fin}^{\otimes k}\). We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these \(\sigma \) -closed forcings and their relationships with the classical pseudointersection number \({\mathfrak {p}}\). (shrink)

In this paper we study abstract elementary classes with Löwenheim-Skolem number $\kappa$ , where $\kappa$ is cofinal with $\omega$ , which have finite character. We generalize results obtained by Kueker for $\kappa=\omega$ . In particular, we show that $\mathbb{K}$ is closed under $L_{\infty,\kappa}$ -elementary equivalence and obtain sufficient conditions for $\mathbb{K}$ to be $L_{\infty,\kappa}$ -axiomatizable. In addition, we provide an example to illustrate that if $\kappa$ is uncountable regular then $\mathbb{K}$ is not closed under $L_{\infty,\kappa}$ -elementary equivalence.

Charles Darwin's Origin of Species, published in 1859, was a revolutionary attempt “to overthrow the dogma of separate creations,” a declaration that provoked different reactions among the religious, ranging from mild enthusiasm to anger. Christians sympathetic to Darwin's effort sought to make Darwinism appear compatible with their religious beliefs. Two of Darwin's most prominent defenders in the United States were the Calvinists Asa Gray, a Harvard botanist, and George Frederick Wright, a cleric-geologist. Gray, who long favored a “special origination” in (...) connection with the evolution of humans and questioned whether natural selection can account for the formation of organs, the making of eyes, etc., embraced natural selection as the primary mechanism underlying the production of most species. He also went so far as to invoke divine providence, rather than randomness, to explain the variations on which natural selection acted. This chapter discusses creationism, intelligent design, and modern biology. (shrink)

We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we characterize the degrees which have a least upper bound with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees (...) are definable in $\operatorname {\mathrm {\mathbf {ER}}}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf {d}\oplus \mathbf {\operatorname {\mathrm {\mathbf {Id}}}_1}$ needn’t be a strong minimal cover of a self-full degree $\mathbf {d}$. Finally, we show that the theory of the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second-order arithmetic. (shrink)

This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L$, JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $\ (...) class='Hi'>omega $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model.In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property.Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show that there are no infinite mad families in $L$ under large cardinals and that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$, respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L$. Part I concludes with Chapter 5, a short list of open questions.In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s.The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T, then $L$ -absoluteness for ccc posets must fail.We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L.In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L$ -absoluteness for $\lambda $ -linked proper posets.Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L.prepared by Zach Norwood.E-mail: [email protected]. (shrink)

As the crusade to outlaw the teaching of evolution changed to a battle for equal time for creationism, the ideological defenses of that doctrine also shifted from primarily biblical to more scientific grounds. This essay describes the historical development of “scientific creationism” from a variety of late–nineteenth– and early–twentieth–century creationist reactions to Charles Darwin's theory of evolution, through the Scopes trial and the 1960s revival of creationism, to the current spread of strict creationism around the world.

Belief in the divine origin of the universe began to wane most markedly in the nineteenth century, when scientific accounts of creation by natural law arose to challenge traditional religious doctrines. Most of the credit - or blame - for the victory of naturalism has generally gone to Charles Darwin and the biologists who formulated theories of organic evolution. Darwinism undoubtedly played the major role, but the supporting parts played by naturalistic cosmogonies should also be acknowledged. Chief among these was (...) the nebular hypothesis proposed by Pierre Simon Laplace in 1796, which explained the origin of the solar system as a natural development over extended periods of time. Ronald Numbers focuses on Laplace's theory as it affected American scientific thought. he first traces the history of Laplace's cosmogony chronologically, from its European inception to its demise about 1900. the last three chapters explore some of the theological and scientific consequences resulting from the acceptance of this cosmogony. Most significant was the change in the status of supernatural doctrine. When the nebular hypothesis lost credence at the end of the nineteenth century, those who had before tried to accommodate natural theory with supernatural doctrine no longer felt compelled to do so when faced with succeeding theories. The nebular hypothesis, it seems, had established natural law in the heavens. (shrink)

In this note, the problem of tracking random references and rejecting random perturbations in a quadrotor, both generated by an auxiliary system named exosystem, is solved by extending the deterministic tracking problem to the area of stochastic processes. Besides, it is considered that only a part of the state vector of the quadrotor is available through measurements. As a consequence, the state vector of the plant must be estimated in order to close the control loop. On this basis, a controller (...) to track random references and to reject random perturbations is developed by combining a Kalman filter to estimate the references and perturbations of an exosystem and an observer to estimate the states of a quadrotor. Besides, to obtain a more practical controller, the analysis is carried out in discrete time. Numerical simulations are used in a quadrotor to confirm the validity and effectiveness of the proposed control. (shrink)

As past president of both the History of Science Society and the American Society of Church History, Ronald L. Numbers is uniquely qualified to assess the historical relations between science and Christianity. In this collection of his most recent essays, he moves beyond the clichés of conflict and harmony to explore the tangled web of historical interactions involving scientific and religious beliefs. In his lead essay he offers an unprecedented overview of the history of science and Christianity from the perspective (...) of the ordinary people who filled the pews of churchesor loitered around outside. Unlike the elite scientists and theologians on whom most historians have focused, these vulgar Christians cared little about the discoveries of Copernicus, Newton, and Einstein. Instead, they worried about the causes of the diseases and disasters that directly affected their lives and about scientists preposterous attempts to trace human ancestry back to apes. Far from dismissing opinion-makers in the pulpit, Numbers closely looks at two the most influential Protestant theologians in nineteenth-century America: Charles Hodge and William Henry Green. Hodge, after decades of struggling to harmonize Gods two revelationsin nature and in the Biblein the end famously described Darwinism as atheism. Green, on the basis of his careful biblical studies, concluded that Ussher's chronology was unreliable, thus opening the door for Christian anthropologists to accommodate the subsequent discovery of human antiquity. In Science without God Numbers traces the millennia-long history of so-called methodological naturalism, the commitment to explaining the natural world without appeals to the supernatural. By the early nineteenth century this practice was becoming the defining characteristic of science; in the late twentieth century it became the central point of attack in the audacious attempt of intelligent designers to redefine science. Numbers ends his reassessment by arguing that although science has markedly changed the world we live in, it has contributed less to secularizing it than many have claimed. Taken together, these accessible and authoritative essays form a perfect introduction to Christian attitudes towards science since the 17th century. (shrink)

Like other English-speaking peoples around the world, New Zealanders began debating Darwinism in the early 1860s, shortly after the publication of Charles Darwin's Origin of Species. Despite the opposition of some religious and political leaders – and even the odd scientist – biological evolution made deep inroads in a culture that increasingly identified itself as secular. The introduction of pro-evolution curricula and radio broadcasts provoked occasional antievolution outbursts, but creationism remained more an object of ridicule than a threat until the (...) last decades of the twentieth century, when first American and then Australian creationists began fomenting antievolutionism among New Zealanders. Although Stephen Jay Gould assured them in 1986 that they had little to fear from so-called scientific creationism, because it was a ‘peculiarly American’ phenomenon, scientific creationism by the mid-1990s had captured the allegiance of an estimated five per cent of the country and proved especially attractive to Maori and Pacific Islanders. In 1992 New Zealand creationists formed their own antievolution society, Creation Science. (shrink)