This paper focuses on four passages in the journey of the universe from beginning to end: its origin in the Big Bang, the production of heavy elements in first generation stars, the buzzing symphony of life on earth, and the distant future of the cosmos. As a physicist and a Christian theologian, I will ask how each of these passages casts light on the deepest questions of existence and our relation to God, and in turn how these questions are being (...) explored through ongoing research into the interaction between Christian theology and the natural sciences. (shrink)

This paper discusses the relation between the minimal positive relevant logic B and intersection and union type theories. There is a marvelous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking BT of B, which saves implication and conjunction but drops disjunction . The filter models of the -calculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models (...) of these calculi. (The logician's "theory" is the algebraist's "filter".) The coincidence extends to a dual interpretation of key particles—the subtype translates to provable , type intersection to conjunction , function space to implication, and whole domain to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper, and to be expected that type union should correspond to the logical disjunction of B. But the simulation of functional application by a fusion (or modus ponens product) operation on theories leaves the key Bubbling lemma of work on ITD unprovable for the -prime theories now appropriate for the modeling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of and CL. (shrink)

Two countable filters on $\omega$ are incompatible if they have no common infinite pseudointersection. Letting $\alpha(P_f)$ denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on $\omega$, we prove the consistency of t $< \alpha(P_f)$.

We study the $I$-ultrafilters on $\omega$, where $I$ is a collection of subsets of a set $X$, usually $\mathbb{R}$ or $\omega_1$. The $I$-ultrafilters usually contain the $P$-points, often as a small proper subset. We study relations between $I$-ultrafilters for various $I$, and closure of $I$-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether $I$-ultrafilters always exist.

Logics $L^F(M)$ are considered, in which $M$ ("most") is a new first-order quantifier whose interpretation depends on a given filter $F$ of subsets of $\omega$. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that $F$ is not of the form $\{(\bigcap F) \cup X: |\omega - X| < \omega\}$ with $|\omega - \bigcap F| = \omega$. Moreover the set of validities of $L^F(M)$ and even of $L^F_{\omega_1\omega}(M)$ depends only on a few basic properties (...) of F. Similar characterizations are given of the class of filters $F$ for which $L^F(M)$ has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models $(\mathfrak{U}, q)$, where the collection $q$ is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter $F$). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity ( $\hbox{\sf KP} \omega$ ) given in [1] uses a choice functional (which is not a definable set function of ( $hbox{\sf KP} \omega$ ). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive recursive in $x\mapsto\omega$ . This yields the following characterization: The class of $\Sigma$ -definable set functions of $\hbox{\sf KP} \omega$ coincides (...) with the collection of set functionals of type 1 primitive recursive in $x\mapsto \omega$. (shrink)

This paper reports on the integration of the higher-order theorem proving environment TPS [Andrews96] into the mathematical assistant OMEGA [Omega97]. TPS can be called from OMEGA either as a black box or as an interactive system. In black box mode, the user has control over the parameters which control proof search in TPS; in interactive mode, all features of the TPS-system are available to the user. If the subproblem which is passed to TPS contains concepts defined in OMEGA’s database of (...) mathematical theories, these definitions are not instantiated but are also passed to TPS. Using a special theory which contains proof tactics that model the ND-calculus variant of TPS within OMEGA, any complete or partial proof generated in TPS can be translated one to one into an OMEGA proof plan. Proof transformation is realised by proof plan expansion in OMEGA’s 3-dimensional proof data structure, and remains transparent to the user. (shrink)

For a free filter F on $$\omega $$ ω, endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F }, where $$p_F\not \in \omega $$ p F ∉ ω, with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F. Spaces of the form $$N_F$$ N F constitute the (...) class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space $$N_F$$ N F carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that $$\mu _n(f)\rightarrow 0$$ μ n ( f ) → 0 for every bounded continuous real-valued function f on $$N_F$$ N F if and only if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, that is, the dual ideal $$F^*$$ F ∗ is Katětov below the asymptotic density ideal $${\mathcal {Z}}$$ Z. Consequently, we get that if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, then: (1) if X is a Tychonoff space and $$N_F$$ N F is homeomorphic to a subspace of X, then the space $$C_p^*(X)$$ C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space $$c_0$$ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$N_F$$ N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. (shrink)

Da Costa's C systems are surveyed and motivated, and significant failings of the systems are indicated. Variations are then made on these systems in an attempt to surmount their defects and limitations. The main system to emerge from this effort, system $CC_{\omega}$ , is investigated in some detail, and "dual-intuitionistic" semantical analyses are developed for it and surrounding systems. These semantics are then adapted for the original C systems, first in a rather unilluminating relational fashion, subsequently in a more illuminating (...) way through the introduction of impossible situations where and and or change roles. Finally other attempts to break out of impasses for the original and expanded C systems, by going inside them, are looked at, and further research directions suggested. (shrink)

In the second chapter to his Galatians letter, Paul makes some striking statements. He says that he has been “crucified with Christ,” and indeed that he no longer lives but that Christ lives “in” him. Such claims raise fascinating exegetical and metaphysical issues that are important for theology. Just who is this “I”, and what is the relation of this “I” to Christ? How are we to understand union with Christ – indeed, is the relation spoken of here something stronger (...) than mere union? Is it identity? In this essay, I offer an analytic engagement with traditional and more recent “apocalyptic” interpretations of this passage, and I argue that a traditional account is preferable. (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)

The subject of this paper is a characterization of the $\Sigma_1$ -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: $KP\omega+(uniform\;AC)$ . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and $x\mapsto\omega$ . This goal is achieved by a Gödel Dialectica-style functional interpretation of $KP\omega+(uniform\;AC)$ and a computability proof for the involved functionals.

The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hbox{\sf KP} \omega$\end{document}) given in [1] uses a choice functional (which is not a definable set function of (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $hbox{\sf KP} \omega$\end{document}). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive recursive in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $x\mapsto\omega$\end{document}. This yields the following characterization: The class of \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}-definable set functions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hbox{\sf KP} \omega$\end{document} coincides with the collection of set functionals of type 1 primitive recursive in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $x\mapsto \omega$\end{document}. (shrink)

The paper "Partitions of Products" [DiPH] investigated the polarized partition relation $\begin{pmatrix}\omega\\\omega\\\omega\\\vdots\end{pmatrix} \rightarrow \begin{pmatrix}\alpha_1\\\alpha_1\\\alpha_2\\\vdots \end{pmatrix}$ The relation is consistent relative to an inaccessible cardinal if every α i is finite, but inconsistent if two are infinite. We show here that it consistent (relative to an inaccessible) for one to be infinite. Along the way, we prove an interesting proposition from ZFC concerning partitions of the finite subsets of ω.

Answering a question of A. Kanamori, we prove that there is no Jonsson filter over the least Jonsson cardinal. We also notice that, assuming $\square_\kappa$, the successor $\kappa^+$ cannot be Jonsson.

In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions (but not over relations). We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for (...) which Heyting-valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes. (shrink)

Jech and Shelah in J Symb Log, 55, 822–830 (1990) studied full reflection below ${\aleph_\omega}$ , and produced a model in which the extent of full reflection is maximal in a certain sense. We produce a model in which full reflection is maximised in a different direction.

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)

Alors que l'eschatologie courante des religions archaïques se situe dans la croyance au retour éternel, au recommencement sans fin d'un monde sans issue, le Dieu d'Israël ouvre un avenir à la foi. Abraham découvre un commencement de l'histoire, un Dieu attentif et plus fort que la mort. Constamment menacé de périr, Israël apprend des prophètes à reconnaître le jugement de Dieu et à reprendre vie dans la foi. À l'horizon d'une histoire de violence, une espérance se profile. L'apocalyptique sera la (...) rencontre du prophétisme avec la sagesse, découverte de l'histoire et de l'univers, attente d'un accomplissement. En parlant ou en faisant parler de lui sous le nom de Fils de l'homme, Jésus définit sa mission d'incarner l'espérance d'Israël étendue à toute l'humanité : en lui l'eschatologie est actuellement présente dans la création de Dieu. While the common eschatology of archaic religions is based on the belief in an eternal return, and endless recommencement of a no-exit world, the God of Israel opens a future to faith. Abraham discovered a beginning to history, an attentive God who is stronger than death. Israel, constantly threatened by destruction, learned from the prophets to recognize the judgement of God and to take up life in faith again. On the horizon of a history of violence, a hope takes form. The apocalyptic will be the encounter of prophetism with wisdom, a discovery of history and the universe, an awaiting for fulfillment. In speaking or in seeing that he is spoken about as Son of Man, Jesus defined his mission to incarnate the hope of Israel, spread to all humanity : in him eschatology is now present in the creation of God. (shrink)

It is consistent that, for every $n \geq 2$, every stationary subset of $\omega_n$ consisting of ordinals of cofinality $\omega_k$, where $k = 0$ or $k \leq n - 3$, reflects fully in the set of ordinals of cofinality $\omega_{n - 1}$. We also show that this result is best possible.

In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions . We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for which Heyting-valued completeness (...) is known. Finally, we relate the logic to locally connected geometric morphisms between toposes. (shrink)

For sentences $\phi$ of $L_{\omega_{1},\omega}$, we investigate the question of absoluteness of $\phi$ having models in uncountable cardinalities. We first observe that having a model in $\aleph_{1}$ is an absolute property, but having a model in $\aleph_{2}$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis context and provide sentences for any $\alpha\in\omega_{1}\setminus\{0,1,\omega\}$ for which the existence of a model in $\aleph_{\alpha}$ is nonabsolute . Finally, we present a complete (...) sentence for which model existence in $\aleph_{3}$ is nonabsolute. (shrink)