In [12] we proved the following isotropy-reflection principle:

Theorem. Let F be a formally real field and let Fp denote its Pythagorean closure. The natural embedding of reduced special groups from Gred(F) into Gred(Fp) = G(FP) induced by the inclusion of fields, reflects isotropy.

Here Gred(F) denotes the reduced special group (with underlying group Ḟ/ΣḞ2) associated to the field F, henceforth assumed formally real; cf. [11], Chapter 1, §3, for details.

The result proved in [12] is, in fact, more general. For example, the Pythagorean closure Fp can be replaced in the statement above by the intersection of all real closures of F (inside a fixed algebraic closure). Similar statements hold, more generally, for all relative Pythagorean closures of F in the sense of Becker [3], Chapter II, §3.

Since the notion of isotropy of a quadratic form can be expressed by a first-order formula in the natural language LSG for special groups (with the coefficients as parameters), this result raises the question whether the embedding ιFFp: Gred(F) ↪ G (Fp) is elementary. Further, since the LSG-formula expressing isotropy is positive-existential, one may also ask whether ιFFp reflects all (closed) formulas ofthat kind with parameters in Gred(F).

In this paper we give a negative answer to the first of these questions, for a vast class of formally real (non-Pythagorean) fields F (Prop. 5.1). This follows from rather general preservation results concerning the “Boolean hull” and the “reduced quotient” operations on special groups.

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify Theorems 1 and 2 of [5], due to the first author.

We define a generic notion of cut that applies to many first-order theories. We prove a generic cut elimination theorem showing that the cut elimination property holds for all theories having a so-called pre-model. As a corollary, we retrieve cut elimination for several axiomatic theories, including Church's simple type theory.

Let be the class of atomic models of a countable first order theory. We prove that if is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity properties, that quasiminimal types have U-rank 1, and that the U-rank of any type is finite in the uncountably categorical, excellent case. However, in contrast to the first order case, the supremum of the U-rank over all types may be ω (and is not achieved). We illustrate the theory with the example of free groups, and Zilber's pseudo analytic structures.

We say that a computably enumerable (c. e.) degree a is plus-cupping, if for every c.e. degree x with 0 < x ≤ a, there is a c. e. degree y ≠ 0′ such that x ∨ y = 0′. We say that a is n-plus-cupping, if for every c. e. degree x, if 0 < x ≤ a, then there is a lown c. e. degree I such that x ∨ I = 0′. Let PC and PCn be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then PC1 ⊆ PC2 ⊆ PC3 = PC. In this paper we show that PC1 ⊂ PC2, so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [14] showing that LC1 ⊂ LC2, as well as extending the Harrington plus-cupping theorem [8].

We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise the revision theoretically definable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta's theory of truth over arithmetic using fully varied revision sequences yields a complete Σ31 set of integers; (iii) the set of stably categorical sentences using their revision operator Ψ is similarly Σ31 and which is complete in GÖdel's universe of constructive sets L; (iv) give an alternative account of a theory of truth—realistic variance that simplifies full variance, whilst at the same time arriving at Kripkean fixed points.

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size c and decreasing chains of length c+ bellow every element. Assuming t = c a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every ωω-bounding forcing ℙ of size c there is a Cohen-destructible, ℙ-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.

We prove that every countable jump upper semilattice can be embedded in , where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D, ≤T, ∨, ′〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1-type of jusl with 0 is realized in . On the other hand, we show that every quantifier free 1-type of jump partial ordering (jpo) with 0 is realized in . Moreover, we show that if every quantifier free type, p(x1,…, xn), of jpo with 0, which contains the formula x1 ≤ 0(m) & … & xn ≤ 0(m) for some m, is realized in , then every quantifier free type of jpo with 0 is realized in .

We also study the question of whether every jusl with the c.p.p. and size is embeddable in . We show that for the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)

We show that g ≤ c(Sym(ω)) where g is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas.

This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(α) on Baire space with the property that every constructive partial functional defined on {α: R(α)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(α) is equivalent in to ∃βA(α, β) for some stable A(α, β) (which belongs to the classical analytical hierarchy).

The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems, is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that preserves the constructive sense of “or” and “there exists.”

It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the question whether the Kolmogorov-Loveland stochastic sequences are closed under selecting sequences by Kolmogorov-Loveland selection rules, i.e., by not necessarily monotonic, partial computable selection rules. The following previously known results are obtained as corollaries. The Mises-Wald-Church stochastic sequences are not closed under computable permutations, hence in particular they form a strict superclass of the class of Kolmogorov-Loveland stochastic sequences. The Kolmogorov-Loveland selection rules are not closed under composition.

We show that if A and are automorphic via Φ then the structures (A) and () are Δ30-isomorphic via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

We prove that any countable simple unidimensional theory T is supersimple, under the additional assumptions that T eliminates hyperimaginaries and that the Dϕ-ranks are finite and definable.