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We develop a new notion of independence (þ-independence, read“thorn”-independence) that arises from a family of ranks suggestedby Scanlon (þ-ranks). We prove that in a large class of theories(including simple theories and o-minimal theories) this notion hasmany of the properties needed for an adequate geometric structure.
We prove that þ-independence agrees with the usual independencenotions in stable, supersimple and o-minimal theories. Furthermore,we give some evidence that the equivalence between forking andþ-forking in simple theories might be closely related to one of themain open conjectures in simplicity theory, the stable forkingconjecture. In particular, we prove that in any simple theory wherethe stable forking conjecture holds, þ-independence and forkingindependence agree.
We investigate the set (ω) of partitions of the naturalnumbers ordered by ≤* where A ≤* B if by gluingfinitely many blocks of A we can get a partition coarser thanB. In particular, we determine the values of a number ofcardinals which are naturally associated with the structure((ω),≥*), in terms of classical cardinal invariants ofthe continuum.
We present a generalisation of the type-theoretic interpretation ofconstructive set theory into Martin-Löf type theory. Thegeneralisation involves replacing Martin-Löf type theory with anew type theory in which logic is treated as primitive instead ofbeing formulated via the propositions-as-types representation. Theoriginal interpretation treated logic in Martin-Löf type theoryvia the propositions-as-types interpretation. The generalisationinvolves replacing Martin-Löf type theory with a new typetheory in which logic is treated as primitive. The primitivetreatment of logic in type theories allows us to studyreinterpretations of logic, such as the double-negation translation.
We show that there is a restriction, or modification of thefinite-variable fragments of First Order Logic in which a weakform of Craig’s Interpolation Theorem holds but a strong form ofthis theorem does not hold. Translating these results intoAlgebraic Logic we obtain a finitely axiomatizable subvariety offinite dimensional Representable Cylindric Algebras that has theStrong Amalgamation Property but does not have theSuperamalgamation Property. This settles a conjecture ofPigozzi .
Effective domain theory is applied to fuzzy logic. The aim is togive suitable notions of semi-decidable and decidable L-subsetand to investigate about the effectiveness of the fuzzy deductionapparatus.
The quantified relevant logic RQ is given a new semantics in whicha formula ∀ x A is true when there is some trueproposition that implies all x-instantiations of A. Formulaeare modelled as functions from variable-assignments topropositions, where a proposition is a set of worlds in a relevantmodel structure. A completeness proof is given for a basicquantificational system QR from which RQ is obtained by adding theaxiom EC of ‘extensional confinement’: ∀ x(A∨B)→(A∨∀ xB), with x not free in A.Validity of EC requires an additional model condition involvingthe boolean difference of propositions. A QR-model falsifying ECis constructed by forming the disjoint union of two naturalarithmetical structures in which negation is interpreted by theminus operation.
We study definability of second order generalized quantifiers on finite structures.Our main result says that for every second order type t there exists a second ordergeneralized quantifier of type t which is not definable in the extension of secondorder logic by all second order generalized quantifiers of types lower than t.
It is shown that coherence conditions for monoidalcategories concerning associativity are analogous to coherenceconditions for symmetric strictly monoidal categories, whereassociativity arrows are identities. Mac Lane’s pentagonalcoherence condition for associativity is decomposed intoconditions concerning commutativity, among which we have acondition analogous to naturality and a degenerate case of Mac Lane’s hexagonal condition for commutativity. This decompositionis analogous to the derivation of the Yang-Baxter equation fromMac Lane’s hexagon and the naturality of commutativity. Thepentagon is reduced to an inductive definition of a kind ofcommutativity.
It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the T-function which is peculiar to NF turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente.
Let N be a transitive model of ZFC such that ω N ⊂ N and 𝒫(ℝ) ⊂ N. Assume that both V and N satisfy “the core model K exists.” Then KN is an iterate of K, i.e.,there exists an iteration tree 𝒯 on K such that 𝒯 has successor length and ℳ𝒯∞ = KN. Moreover, if there exists an elementary embedding π : V → N then the iteration map associated to the main branch of 𝒯 equals π ↾ K. (This answers a questionof W. H. Woodin, M. Gitik, and others.) The hypothesis that 𝒫(ℝ) ⊂ N is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals.
In , Yates proved the existence of a Turing degree asuch that 0, 0’ are the only c.e. degrees comparable with it.By Slaman and Steel , every degree below 0’ has a 1-genericcomplement, and as a consequence, Yates degrees can be 1-generic,and hence can be low. In this paper, we prove that Yates degrees occur in every jump class.
In this paper we show that there are “E0 many” orbitinequivalent free actions of the free groups 𝔽n, 2≮n≮∞ by measure preserving transformations on a standardBorel probability space. In particular, there are uncountably manysuch actions.
Makkai  produced an arithmetical structure of Scott rank ω1CK.In , Makkai’s example is made computable. Here we show that there arecomputable trees of Scott rank ω1CK. We introduce a notion of “rankhomogeneity”. In rank homogeneous trees, orbits of tuples can be understoodrelatively easily. By using these trees, we avoid the need to pass to the morecomplicated “group trees” of  and . Using the same kind oftrees, we obtain one of rank ω1CK that is “strongly computablyapproximable”. We also develop some technology that may yield further resultsof this kind.
Nominal logic is a variant of first-order logic in which abstractsyntax with names and binding is formalized in terms of two basicoperations: name-swapping and freshness. It relieson two important principles: equivariance (validity ispreserved by name-swapping), and fresh name generation(“new” or fresh names can always be chosen). It is inspired by aparticular class of models for abstract syntax trees involving namesand binding, drawing on ideas from Fraenkel-Mostowski set theory:finite-support models in which each value can depend on onlyfinitely many names.
Although nominal logic is sound with respect to such models, it isnot complete. In this paper we review nominal logic and show why finite-support models are insufficient both in theory and practice. We then identify (up to isomorphism) the class of models with respect to which nominal logic is complete: ideal-supported models in which the supports of values are elements of a proper ideal on the set of names.
We also investigate an appropriate generalization of Herbrand modelsto nominal logic. After adjusting the syntax of nominal logic toinclude constants denoting names, we generalize universaltheories to nominal-universal theories and prove that eachsuch theory has an Herbrand model.
A structure of finite signature is constructed so that: for allexistential formulas ∃y⃗ φ(x⃗,y⃗) andfor all tuples of elements ⃗ of the same length as the tuplex⃗, one can decide in a quadratic time depending only on thelength of the formula, if ∃y⃗ φ(u⃗,y⃗)holds in the structure. In other words, the structure satisfies therelativized model-theoretic version of P=NP in the sense of. This is a model-theoretical approach to results ofHemmerling and Gaßner.
We define a notion of realizability, based on a new assignment offormulas, which does not care for precise witnesses of existentialstatements, but only for bounds for them. The novel form ofrealizability supports a very general form of the FAN theorem, refutesMarkov’s principle but meshes well with some classicalprinciples, including the lesser limited principle of omniscience andweak König’s lemma. We discuss some applications, aswell as some previous results in the literature.
We investigate the geometry of forking for SU-rank 2 elements insupersimple ω-categorical theories and prove stable forking andsome structural properties for such elements. We extend this analysisto the case of SU-rank 3 elements.
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