We show that every nonzero $${\Delta^{0}_{2}}$$ enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper.

Say a set Gω is 1-generic if for any eω, there is a string σG such that {e}σ↓ or τσ↑). It is known that can be split into two 1-generic degrees. In this paper, we generalize this and prove that any nonzero computably enumerable degree can be split into two 1-generic degrees. As a corollary, no two computably enumerable degrees bound the same class of 1-generic degrees.

We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Δ such that, for any A, the set ΔA is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such that a'=b', but also (...) that, if a is nonzero, then we can find such b satisfying 0eshrink)

There is a family of questions in relativized complexity theory--weak analogs of the Friedberg Jump-Inversion Theorem--that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2-generic sets. i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it (...) is shown that for any set X there is an aw2-generic set G such that $\mathbf{NP}^G \cap co-\mathbf{NP}^G \nsubseteq \mathbf{P}^{G \oplus X}$ . (On the other hand, if G is 1-generic, then $\mathbf{NP}^G \cap co-\mathbf{NP}^G \subseteq \mathbf{P}^{G \oplus \mathrm{SAT}}$ . where SAT is the NP-complete satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2-generic sets differ radically from those of 1-generic sets: every degree above O' contains an aw2-generic set: no aw2-generic set exists below any incomplete r.e. set; there is an aw2-generic set which is the join of two Turing equivalent aw2-generic sets. Finally, a result of Shore is presented [30] which states that every degree above 0' is the jump of an aw2-generic degree. (shrink)

A set of natural numbers B is computably enumerable in and strictly above (or c.e.a. for short) another set C if C < T B and B is computably enumerable in C. A Turing degree b is c.e.a. c if b and c respectively contain B and C as above. In this paper, it is shown that if b is c.e.a. c then b is c.e.a. some 1-generic g.

We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of reals in (...) models of ZF. In particular, we show that ZFC + is equiconsistent with ZF + \\) where \ is the pointclass of all \-Suslin sets of reals, and also with ZF + \\) + \\) where \ is the least ordinal that is not a surjective image of the reals. (shrink)

A generic computation of a subsetAof ℕ is a computation which correctly computes most of the bits ofA, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory, where it has been noticed that frequently, it is more important to know how difficult a type of problem is in the general case than how difficult it is in the worst case. When we study this concept from a recursion theoretic point of (...) view, to create a transitive relationship, we are forced to consider oracles that sometimes fail to give answers when asked questions. Unfortunately, this makes working in the generic degrees quite difficult. Indeed, we show that generic reduction is$\Pi _1^1$―complete. To help avoid this difficulty, we work with the generic degrees of density-1 reals. We demonstrate how an understanding of these degrees leads to a greater understanding of the overall structure of the generic degrees, and we also use these density-1 sets to provide a new a characterization of the hyperartithmetical Turing degrees. (shrink)

It is natural to wish to study miniaturisations of Cohen forcing suitable to sets of low arithmetic complexity. We consider extensions of the work of Schaeffer [9] and Jockusch and Posner [6] by looking at genericity notions within the Δ2 sets. Different equivalent characterisations of 1-genericity suggest different ways in which the definition might be generalised. There are two natural ways of casting the notion of 1-genericity: in terms of sets of strings and in terms of density (...) functions, as we will see here. While these definitions coincide at the first level of the difference hierarchy, they turn out to differ at other levels. Furthermore, these differences remain when the remainder of the Δ02 sets are considered. While the string characterization of 1-genericity collapses at the second level of the difference hierarchy to 2-genericity, the density function definition gives a very interesting hierarchy at level w and above. Both of these results point towards the deep similarities exhibited by the n-c.e. degrees for n ≥ 2. (shrink)

This paper investigates indifferent sets for comeager classes in Cantor space focusing of the class of all 1-generic sets and the class of all weakly 1-generic sets. Jockusch and Posner showed that there exist 1-generic sets that have indifferent sets [10]. Figueira, Miller and Nies have studied indifferent sets for randomness and other notions [7]. We show that any comeager class in Cantor space contains a comeager class with a universal indifferent set. A forcing (...) construction is used to show that any 1-generic set, or weakly 1-generic set, has an indifferent set. Such an indifferent set can by computed by any set in GL2 which bounds the (weakly) 1-generic. We show by approximation arguments that some, but not all, ∆0 1-generic sets can compute an indifferent set 2 for themselves. We show that all ∆0 weakly 1-generic sets can compute 2 an indifferent set for themselves. Additional results on indifferent sets, including one of Miller, and two of Fitzgerald, are presented. (shrink)

For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T . We show that the theory T* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T* coincides with the theory of infinite dimensional pairs, which was used in 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T* for the same purpose. In particular, we obtain a characterization of linearity (...) for SU-rank 1 structures by giving several equivalent conditions on T*, find a “weak” version of local modularity which is equivalent to linearity, show that linearity coincides with 1-basedness, and use the generic pairs to “recover” projective geometries over division rings from non-trivial linear SU-rank 1 structures. (shrink)

We study the degrees of unsolvability of sets which are cohesive . We answer a question raised by the first author in 1972 by showing that there is a cohesive set A whose degree a satisfies a' = 0″ and hence is not high. We characterize the jumps of the degrees of r-cohesive sets, and we show that the degrees of r-cohesive sets coincide with those of the cohesive sets. We obtain analogous results for strongly hyperimmune (...) and strongly hyperhyperimmune sets in place of r-cohesive and cohesive sets, respectively. We show that every strongly hyperimmune set whose degree contains either a Boolean combination of ∑2 sets or a 1-generic set is of high degree. We also study primitive recursive analogues of these notions and in this case we characterize the corresponding degrees exactly. MSC: 03D30, 03D55. (shrink)

We say that a real X is n-generic relative to a perfect tree T if X is a path through T and for all $\Sigma _{n}^{0}(T)$ sets S, there exists a number k such that either X|k ∈ S or for all σ ∈ T extending X|k we have σ ∉ S. A real X is n-generic relative to some perfect tree if there exists such a T. We first show that for every number n all but countably many (...) reals are n-generic relative to some perfect tree. Second, we show that proving this statement requires ZFC− + "∃ infinitely many iterates of the power set of ω". Third, we prove that every finite iterate of the hyperjump. ${\cal O}^{(n)}$ , is not 2-generic relative to any perfect tree and for every ordinal α below the least λ such that supβ<i (βth admissible) = λ, the iterated hyperjump ${\cal O}^{(\alpha)}$ is not 5-generic relative to any perfect tree. Finally, we demonstrate some necessary conditions for reals to be 1-generic relative to some perfect tree. (shrink)

There is a comeager set C contained in the set of 1-generic reals and a first order structure M such that for any real number X, there is an element of C which is recursive in X if and only if there is a presentation of M which is recursive in X.

While opinions on the semantic analysis of generics vary widely, most scholars agree that generics have a quasi-universal flavor. However, there are cases where generics receive what appears to be an existentialinterpretation. For example, B's response is true, even though only theplatypus and the echidna lay eggs: (1) A: Birds lay eggs. B: Mammals lay eggs too. In this paper I propose a uniform account of the semantics of generics,which accounts for their quasi-existential readings as well as for their more (...) familiar quasi-universal ones. Generics are focus-sensitiveoperators: their domain is restricted by a set of alternatives, which may be provided by focus. I claim that, unlike otherfocus-sensitive operators, generics may, but do not have to, associate with focus. When alternatives are introduced, either by focus or by other means, generics get their usual quasi-universal readings. But when no alternatives are introduced, quasi-existential readings result.I argue that generics, unlike adverbs of quantification, do not introduce tripartite structures directly, but are initially interpreted as cases ofdirect kind predication. Only when this interpretation fails to make sense, the phonologically null generic quantifier is derived, and tripartite structures result. This two-level interpretation has the effect that while adverbs of quantification require focus to determine which elements go to the restrictor and which to the nuclear scope, and hence must associate with focus, generics do not, and hence may fail to associate with focus, resulting in quasi-existential readings. (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra (...) set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true , etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)

A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set _A_ in this ultrafilter such that the restriction of the function to _A_ is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\). (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) such that every filter base (...) of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\). (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\). (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra (...) set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)

LetMbe a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by definingd(g, h) = Ω{2−n:g(xn) ≠h(xn) org−1(xn) ≠h−1(xn)} where {xn:n∈ ω} is an enumeration ofMAn automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In the (...) present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms ofMwhich can be considered as an atomic model of some theory naturally associated toM. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks. (shrink)

While finitely generic dimension groups are known to admit no proper self-embeddings, these groups also have no automorphisms other than scalar multiplications, and every countable infinitely generic dimension group admits proper self-embeddings and has automorphisms other than scalar multiplications. The finite-forcing companion of the theory of dimension groups is recursively isomorphic to first-order arithmetic, the infinite-forcing companion of the theory of dimension groups is recursively isomorphic to second-order arithmetic, and the first-order theory of existentially closed dimension groups is a complete (...) $\Pi_{1}^{1}$-set. While many special properties of f.g. dimension groups may be realized in recursive e.c. dimension groups, and many special properties of i.g. dimension groups may be realized in hyperarithmetic e.c. dimension groups, no f.g. dimension group is arithmetic and no i.g. dimension group is analytical. Yet there is an f.g. dimension group recursive in first-order arithmetic, and there is an i.g. dimension group recursive in second-order arithmetic. (shrink)

In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees.

We consider questions related to the rigidity of the structure R, the PTIME-Turing degrees of recursive sets of strings together with PTIME-Turing reducibility, pT, and related structures; do these structures have nontrivial automorphisms? We prove that there is a nontrivial automorphism of an ideal of R. This can be rephrased in terms of partial relativizations. We consider the sets which are PTIME-Turing computable from a set A, and call this class PTIMEA. Our result can be stated as follows: (...) There is an oracle, A, relative to which the PTIME-Turing degrees are not rigid . Furthermore, the automorphism can be made to preserve the complexity classes DTIMEA for all k 1, or to move any DTIMEA for n 2. From the existence of such an automorphism we conclude as a corollary that there is an oracle A relative to which the classes DTIME are not definable over R. We carry out the corresponding partial relativization for the many-one degrees to construct an oracle, A, relative to which the PTIMA-many-one degrees relative to A have a nontrivial automorphism, and one relative to which the lattice of sets in PTIMEA under inclusion have a nontrivial automorphism. The proof is phrased as a forcing argument; we construct the set A to meet a particular collection of dense sets in our notion of forcing. Roughly, the dense sets will guarantee that if A meets these sets and we split A into two pieces, A0 and A1, in a simple way, and then switching the roles of A0 and A1 in all computations from A will produce an automorphism of the ideal of PTIMA-degrees below A. We force A0 and A1 to have different PTIME-Turing degree; this will then make the automorphism nontrivial. An appropriately generic set A is constructed using a priority argument. (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra (...) set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)

Children and adults commonly produce more generic noun phrases (e.g., birds fly) about animals than artifacts. This may reflect differences in participants’ generic knowledge about specific animals/artifacts (e.g., dogs/chairs), or it may reflect a more general distinction. To test this, the current experiments asked adults and preschoolers to generate properties about novel animals and artifacts (Experiment 1: real animals/artifacts; Experiments 2 and 3: matched pairs of maximally similar, novel animals/artifacts). Data demonstrate that even without prior knowledge about these items, the (...) likelihood of producing a generic is significantly greater for animals than artifacts. These results leave open the question of whether this pattern is the product of experience and learned associations or instead a set of early-developing theories about animals and artifacts. (shrink)

Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James (...) H. Schmerl; 9. History of constructivism in the 20th century A. S. Troelstra; 10. A very short history of ultrafinitism Rose M. Cherubin and Mirco A. Mannucci; 11. Sue Toledo's notes of her conversations with Gödel in 1972-1975 Sue Toledo; 12. Stanley Tennenbaum's Socrates Curtis Franks; 13. Tennenbaum's proof of the irrationality of [the square root of] 2́. (shrink)

If A ⊆ ω1, then there exists a cardinal preserving generic extension [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ][x ] of [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ] by a real x such that1) A ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ] and A is Δ1HC in [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ];2) x is minimal over [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ], that is, if a set Y belongs to [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ], then either x ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A, Y ] or Y (...) ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ].The forcing we use implicitly provides reshaping of the given set A. (shrink)

We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is forceable, then (...) it is true. Further, if for every real x, $x^{\sharp}$ exists, and the second uniform indiscernible is less than $\omega_2$ , then the same holds for $\Sigma^1_4$ sentences. (shrink)

Causal sets are a particular class of partially ordered sets, which are proposed as basic models of discrete space-time, specially in the field of quantum gravity. In this context, we show the existence of temporal foliations for any causal set, or more generally, for a causal space. Moreover, we show that automorphisms of a large class of infinite causal sets fall into two classes 1) Automorphisms of spacelike hypersurfaces in some given foliation, or 2) Translations in time. (...) More generally, we show that for any automorphism \ of a generic causal set \, there exists a partition of \ into finitely many subcausets, on each of which or above hold. These subcausets can be assumed connected if, in addition, there are enough distinct orbits under \. (shrink)

A DO model (here also referred to a Paris model) is a model of set theory all of whose ordinals are first order definable in . Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include the (...) following:1. If T is a consistent completion of ZF+V≠OD, then T has continuum-many countable nonisomorphic Paris models.2. Every countable model of ZFC has a Paris generic extension.3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism.4. For a model ZF, is a prime model ⇒ is a Paris model and satisfies AC⇒ is a minimal model. Moreover, Neither implication reverses assuming Con(ZF). (shrink)

In lieu of an abstract, here is a brief excerpt of the content:That St(r)ain Again:Blood, Water, and Generic Allusion in Horace's Bandusia OdeGottfried MaderAbstractHorace's vivid picture of the blood sacrifice to the spring of Bandusia has left many readers feeling somewhat uneasy, for while animal sacrifices appear elsewhere in the Odes,1 none matches this for its pathos or detail:O fons Bandusiae, splendidior vitro,dulci digne mero non sine floribus, cras donaberis haedo, cui frons turgida cornibusprimis et venerem et proelia destinat.frustra: nam (...) gelidos inficiet tibi rubro sanguine rivos lascivi suboles gregis.(Carm. 3.13.1-8)O spring of Bandusia, brighter than glass, worthy of sweet wine and flowers: tomorrow you will be presented with a kid whose forehead, swelling with first horns, destines him for both love and battles—but in vain, for to honour you the offspring of the wanton herd will dye your icy waters with his crimson blood.Hence the protest or surprise expressed at this allegedly offensive intrusion of blood into an otherwise idyllic scene: "Who wants a drink out of the fountain of Bandusia after that?" (Campbell 1924, 2); "[Horace] visualizes with aesthetic relish the mingling of the cool water and the red blood.... [His] behaviour was strange even by ancient criteria" (Nisbet 1962, 198); "somewhat macabre" (Williams 1969, 89); "the scandal of the sacrificial kid, whose death is dwelt upon... with a cruel brilliance quite out of proportion to any decorative needs" (Wilson 1968, 289).2 Others [End Page 51] have attempted to domesticate the sacrifice, urging that the ancients were less sentimental in these matters and justifying the description as rustic realism.3 But neither view accounts adequately for the densely suggestive detail or the perceptible emphasis on the blood offering—in a word, for the poetic intensity of these lines.The sacrifice begins to make sense when we explore its metaphorical possibilities and integrate it into the ode's thematic and intellectual design.4 Recent criticism has emphasized the metapoetic dimension of 3.13, interpreting Horace's homage to the spring as a tribute to the poetry which secures it a place among the celebrated fountains of Greek and Roman literature: "fies nobilium tu quoque fontium / me dicente cavis inpositam ilicem / saxis, unde loquaces / lymphae desiliunt tuae " (13-16; "you too will take your place among the famous springs when I celebrate the ilex set upon the hollow rocks, from which your clear-voiced waters come cascading down"). Further, since springs and water have a long metaphorical association with poetry (cf. below, note 12), the literal shades off easily into the literary, and the fons reflects the ideals and aspirations of Horace's own lyric art.5 From this perspective, the blood sacrifice too acquires a metaphorical nuance. Here is Steele Commager: "Readers have often been repelled by the details of the kid's sacrifice (6-8). Perhaps the description is not there for its realistic effect alone. Destined for love and battle, the 'offspring of the wanton flock' epitomizes life's comprehensive vitality, and as his warm blood mingles with the lucid water it is easy to sense a suggestion of the transformation of life into art" (Commager 1962, 323-24). Ralph Hexter continues on this trajectory: "The image of the red blood of the goat staining the cold spring water reflects the transmutation of life into poetry.... I believe a close reading of the poem supports a further insight: that Horace means to show us how life is transferred to the water of poetry, and by showing us how it is, he shows us that it is."6 Most recently, Gregson Davis has suggested a further association: "Unmixed wine, flowers, and young animal constitute the sine quibus non of the banqueting apparatus.... Horace's fons is to be honored with the irreducible tokens of convivial [End Page 52] poetry. This ensemble of tokens, and not the presumed reference to an obscure festival, is what principally determines the speaker's choice of offerings." And once the choice of offerings has established the lyric orientation, the goat's death suggests to Davis also the mortality topos or the "dark background," typically a foil to the convivial motifs in the context of... (shrink)

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega}$. $\mathsf{PROVI}$ supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and Shoenfield’s (...) unramified forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and the extension of a provident $M$ by a set-generic $\mathcal{G}$ is the provident closure of $M\cup\{\mathcal{G}\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$. (shrink)

We present the axioms of Alternative Set Theory in the language of second-order arithmetic and study its ω- and β-models. These are expansions of the form , M ⊆ P, of nonstandard models M of Peano arithmetic such that ⊩ AST and ω ϵ M. Our main results are: A countable M ⊩ PA is β-expandable iff there is a regular well-ordering for M. Every countable β-model can be elementarily extended to an ω-model which is not a β-model. The Ω-orderings (...) of an ω-model are absolute well-orderings iff the standard system SS of M is a β-model of A−2. There are ω-expandable models M such that no ω-expansion of M contains absolute Ω-orderings. There are s-expandable models which are not β-expandable. For every countable β-expansion M of M, there is a generic extension M[G] which is also a β-expansion of M. If M is countable and β-expandable, then there are regular orderings <1, <2 such that neither <1 belongs to the ramified analytical hierarchy of the structure , nor <2 to that of . The result can be improved as follows: A countable M ⊩ PA is β-expandable iff there is a semi-regular well-ordering for M. (shrink)

We show that if I is a precipitous ideal on ω₁ and if θ > ω₁ is a regular cardinal, then there is a forcing P = P(I, θ) which preserves the stationarity of all I-positive sets such that in $V^P $ , is a generic iterate of a countable structure . This shows that if the nonstationary ideal on ω₁ is precipitous and $H_\theta ^\# $ exists, then there is a stationary set preserving forcing which increases $\delta _2^1 (...) $ · Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω₁ is precipitous, then $\delta _2^1 $ = u₂ = ω₂. (shrink)

We show it is consistent that there is a Souslin tree S such that after forcing with S, S is Kurepa and for all clubs $$C \subset \omega _1$$ C ⊂ ω 1, $$S\upharpoonright C$$ S ↾ C is rigid. This answers the questions in Fuchs (Arch Math Logic 52(1–2):47–66, 2013). Moreover, we show it is consistent with $$\diamondsuit $$ ♢ that for every Souslin tree T there is a dense $$X \subseteq T$$ X ⊆ T which does not contain (...) a copy of T. This is related to a question due to Baumgartner in Baumgartner (Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., Reidel, Dordrecht-Boston, pp 239–277, 1982). (shrink)

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. We also (...) show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ℵ2 and arrange in the generic extension that simultaneous reflection holds at ℵ2, and at the same time, every stationary subset of ℵ3 concentrating on points of cofinality ω has a reflection point of cofinality ω1. (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 (...) $\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)

Let κ and λ be infinite cardinals, F a filter on κ, and G a set of functions from κ to κ. The filter F is generated by G if F consists of those subsets of κ which contain the range of some element of G. The set G is $ -closed if it is closed in the $ -topology on κ κ. (In general, the $ -topology on IA has basic open sets all Π i∈ I U i (...) such that, for all $i \in I, U_i \subseteq A$ and $|\{i \in I: U_i \neq A\}| .) The primary question considered in this paper asks "Is there a uniform ultrafilter on κ which is generated by a closed set of functions?" (Closed means $ -closed.) We also establish the independence of two related questions. One is due to Carlson: "Does there exist a regular cardinal κ and a subtree T of $^{ such that the set of branches of T generates a uniform ultrafilter on κ?"; and the other is due to Pouzet: "For all regular cardinals κ, is it true that no uniform ultrafilter on κ is analytic?" We show that if κ is a singular, strong limit cardinal, then there is a uniform ultrafilter on κ which is generated by a closed set of increasing functions. Also, from the consistency of an almost huge cardinal, we get the consistency of CH + "There is a uniform ultrafilter on ℵ 1 which is generated by a closed set of increasing functions". In contrast with the above results, we show that if κ is any cardinal, λ is a regular cardinal less than or equal to κ and P is the forcing notion for adding at least $(\kappa^{ generic subsets of λ, then in V P , no uniform ultrafilter on κ is generated by a closed set of functions. (shrink)

As the first poem of the last book of Ovid'sAmores, 3.1 parallels the programmaticrecusatioof the first two books, which present the traditional opposition of elegy to epic. InAmores3.1, the personified Elegy and Tragedy compete for Ovid's poetic attention, and scholars have accordingly scrutinized the generic tension between elegy and tragedy in this poem. My study, by contrast, focusses on the import of the metapoeticlocusin which Ovid sets his contest between the two genres, by considering the linguistic and allusive play (...) in the opening lines. Ovid exploits the metaphor of literary tradition as an ancient and sacred forest to transform an author's choice of poetic genre into a walk in the woods. Moreover, allusions to Virgil'sAeneid6.179 and Ennius'Annales175 (Sk.) in the first line guide Ovid's audience to expect the more traditional opposition of elegy and epic. The less conventional contest between the genres of elegy and tragedy soon overturns this expectation; nevertheless, elegy's customary opposite, epic, maintains a presence in the form of a woodland context for Ovid's innovative generic opposition. (shrink)

For [Formula: see text], the coarse similarity class of A, denoted by [Formula: see text], is the set of all [Formula: see text] such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric [Formula: see text] on the space [Formula: see text] of coarse similarity classes defined by letting [Formula: see text] be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes (...) under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form [Formula: see text] where [Formula: see text] is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of [Formula: see text]. We then define a distance between Turing degrees based on Hausdorff distance in the metric space [Formula: see text]. We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, [Formula: see text], and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance [Formula: see text] from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability. Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of [Formula: see text] from the perspectives of computability theory and reverse mathematics. (shrink)

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is (...) forcible from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

We prove that if I is a partially ordered set in a countable transitive model M of ZFC then M can be extended by a generic sequence of reals a i , i ∈ I, such that ℵ M 1 is preserved and every a i is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j: j . The structure of the degrees of M-constructibility of reals in the extension is investigated. As applications of the methods involved, we define a cardinal invariant to distinguish (...) product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω 2 -iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds. (shrink)

Posner [6] has shown, by a nonuniform proof, that every ▵ 0 2 degree has a complement below 0'. We show that a 1-generic complement for each ▵ 0 2 set of degree between 0 and 0' can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above $\varnothing'$ . In the second half of the paper, we show that the complementation (...) of the degrees below 0' does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems. (shrink)

The naïve idea of “size” for collections seems to obey both Aristotle’s Principle: “the whole is greater than its parts” and Cantor’s Principle: “1-to-1 correspondences preserve size”. Notoriously, Aristotle’s and Cantor’s principles are incompatible for infinite collections. Cantor’s theory of cardinalities weakens the former principle to “the part is not greater than the whole”, but the outcoming cardinal arithmetic is very unusual. It does not allow for inverse operations, and so there is no direct way of introducing infinitesimal numbers. Here (...) we maintain Aristotle’s principle, instead halving Cantor’s principle to “equinumerous collections are in 1–1 correspondence”. In this way we obtain a very nice arithmetic: in fact, our “numerosities” may be taken to be nonstandard integers. These numerosities appear naturally suited to sets of ordinals, but they depend, for generic sets, on a “labelling” of the universe by ordinals. The problem of finding a canonical way of attaching numerosities to all sets seems to be worth further investigation. (shrink)

We define the notion of a completely determined Borel code in reverse mathematics, and consider the principle $CD - PB$, which states that every completely determined Borel set has the property of Baire. We show that this principle is strictly weaker than $AT{R_0}$. Any ω-model of $CD - PB$ must be closed under hyperarithmetic reduction, but $CD - PB$ is not a theory of hyperarithmetic analysis. We show that whenever $M \subseteq {2^\omega }$ is the second-order part of an ω-model (...) of $CD - PB$, then for every $Z \in M$, there is a $G \in M$ such that G is ${\rm{\Delta }}_1^1$-generic relative to Z. (shrink)

The new field of experimental philosophy has emerged as the methods of psychological science have been brought to bear on traditional philosophical issues. Oxford Studies in Experimental Philosophy will be the place to go to see outstanding new work in the field. It will feature papers by philosophers, papers by psychologists, and papers co-authored by people in both disciplines. The series heralds the emergence of a truly interdisciplinary field in which people from different disciplines are working together to address a (...) shared set of questions.The inaugural volume is roughly structured into four sections. The first three papers focus on recent developments in moral psychology, a topic that has seen lively debate and a great deal of progress over the last decade. The second section highlights three contributions that bring new methods to moral psychology: formal modeling and special populations. The third section brings together four papers that adopt an experimental philosophy approach to novel topics, including intuitive dualism, generics, joint action, and happiness. And the last two papers provide critical and historical context to the development of experimental philosophy. (shrink)

A real is called properly n-generic if it is n-generic but not n+1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.

We show how Abstract Complexity Theory is related to the degrees of unsolvability and develop machinery by which computability theoretic hierarchies with a complexity theoretic flavor can be defined and investigated. This machinery is used to prove results both on hierarchies of Δ 2 0 sets and hierarchies of Δ 2 0 degrees. We prove a near-optimal lower bound on the effectivity of the Low Basis Theorem and a result showing that array computable c.e. degrees are, in some sense, (...) the simplest possible Δ 2 0 degrees. We also examine the growth rates of iterates of m K . Finally, we indicate how complexity theory can be used to analyze notions of genericity intermediate between 1 -genericity and 2 -genericity, and produce a hierarchy of such notions. (shrink)

The tower number ${\mathfrak t}$ and the ultrafilter number $\mathfrak {u}$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of $\omega $ and the almost inclusion relation $\subseteq ^*$ between such subsets. We consider analogs of these cardinal characteristics in computability theory.We say that a sequence $(G_n)_{n \in {\mathbb N}}$ of computable sets is a tower if $G_0 = {\mathbb N}$, $G_{n+1} \subseteq ^* G_n$, and $G_n\smallsetminus G_{n+1}$ is infinite for each (...) n. A tower is maximal if there is no infinite computable set contained in all $G_n$. A tower ${\left \langle {G_n}\right \rangle }_{n\in \omega }$ is an ultrafilter base if for each computable R, there is n such that $G_n \subseteq ^* R$ or $G_n \subseteq ^* \overline R$ ; this property implies maximality of the tower. A sequence $(G_n)_{n \in {\mathbb N}}$ of sets can be encoded as the “columns” of a set $G\subseteq \mathbb N$. Our analogs of ${\mathfrak t}$ and ${\mathfrak u}$ are the mass problems of sets encoding maximal towers, and of sets encoding towers that are ultrafilter bases, respectively. The relative position of a cardinal characteristic broadly corresponds to the relative computational complexity of the mass problem. We use Medvedev reducibility to formalize relative computational complexity, and thus to compare such mass problems to known ones.We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin’s characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e. set computes a maximal tower but no 1-generic $\Delta ^0_2$ -set does so.We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively. (shrink)