Let B be the random real forcing. Miller [Mi] asked if there are ZFC models M ⊆ N such that forcing with BM over N adds a dominating real. A YES answer was provided by Judah and Shelah in [JS], where in a long and sophisticated construction they built such models. In this paper we prove that forcing with BV over VI, where I is the infinitely often equal real forcing of [Mi], adds a dominating real over VI. This greatly simplifies the YES answer to Miller's question. Moreover it turns out that B may be replaced here by E, the eventually different real forcing of [Mi]. This answers the second part of Miller's question. We also prove that both side by side products I × B and I × E add a Hechler dominating real over V.

In this section we establish the main result of the paper; namely, we prove that forcing over VI with either of the posets BV or EV adds a dominating real over VI.

First we recall the definitions of I and E from [Mi]. The infinitely often equal real forcing is the set

ordered by extension. Miller [Mi] proves that I is ωω bounding.

A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory.

We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$ -destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$ -positive set which has finite intersection with every $A\in \mathcal {I}\cap V$ . Also, we discuss the associated variants

$$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y|<\omega\big\},\\ \mathrm{cov}^*(\mathcal{I},+)=&\min\big\{|\mathcal{C}|:\mathcal{C}\subseteq\mathcal{I},\; \forall\;Y\in\mathcal{I}^+\;\exists\;C\in\mathcal{C}\;|Y\cap C|=\omega\big\} \end{align*} $$

Among other results, (1) we give a simple combinatorial characterisation when a real forcing $\mathbb {P}_I$ can $+$ -destroy a Borel ideal $\mathcal {J}$ ; (2) we discuss many classical examples of Borel ideals, their $+$ -destructibility, and cardinal invariants; (3) we show that the Mathias–Prikry, $\mathbb {M}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$ iff $\mathbb {M}(\mathcal {I}^*)\ +$ -destroys $\mathcal {I}$ iff $\mathcal {I}$ can be $+$ -destroyed iff $\mathrm {cov}^*(\mathcal {I},+)>\omega $ ; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$ , and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$ -destroys $\mathcal {I}$ ; and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.

Matthias Schröder has asked the question whether there is a weakest discontinuous problem in the topological version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder’s question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work in Zermelo–Fraenkel set theory with dependent choice and the axiom of determinacy $\mathsf {AD}$. On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for Player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for Player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy $\mathsf {AD}$, but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.

We give four examples of theories in which Kreisel's Conjecture is false: (1) the theory PA(-) obtained by adding a function symbol minus, ‘—’, to the language of PA, and the axiom ∀x∀y∀z (x − y = z) ≡ (x = y + z ∨ (x < y ∧ z = 0)); (2) the theory L of integers; (3) the theory PA(q) obtained by adding a function symbol q (of arity ≥ 1) to PA, assuming nothing about q; (4) the theory PA(N) containing a unary predicate N(x) meaning ‘x is a natural number’. In Section 6 we suggest a counterexample to the so called Sharpened Kreisel's Conjecture.

In this paper we give a positive answer to Julia Robinson's question whether the definability of + and · from S and ∣ that she proved in the case of positive integers is extendible to arbitrary integers (cf. [JR, p. 102]).

The following rough summary is intended to give the non-specialist in fine-structure an idea of what core models are and what they are used for. References are usually given to proofs but very few proofs are given. Also attribution of results is rather careless: I hope to give full historical notes in a forthcoming more detailed exposition.

Jensen is certainly responsible for the bulk of these results. This paper is based in part on a lecture he gave in Oxford in 1979, and my warmest thanks are due to him for his patient explanations of these results. (I should add that he has not seen this paper in proof and should not be held responsible for any false claims which it contains.) Mitchell has been the main driving force behind the generalisations of K, and his influence on the results in §5 is perhaps not adequately reflected there. The language of “hypermeasures” has been replaced here by “extenders”, but the reader of [14] will find translation easy.

Graeme Forbes, Robin Gandy, Wilfrid Hodges and the referee read the paper in proof and I am most grateful for their many suggested improvements.

In this paper we study the notion of $C^{(n)}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{(1)}$ -supercompact but also that the least supercompact is $C^{(1)}$ -supercompact (and even $C^{(n)}$ -supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.

We prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set , where H is an infinite integer, cannot be expressed as a countable union of countably determined sets each of which is U-meager for some cut U with min{cf(U), ci(U)} ≥ ω1. Also, every Borel, or countably determined set C which is U-meager for every cut U is a countable union of Borel, or countably determined sets respectively, which are U-nowhere dense for every cut U. Further, the class of Borel U-meager sets for min{cf(U),ci(U)} ≥ ω1 coincides with the least family of sets containing internal U-meager sets and closed with respect to the operation of countable union and intersection. The same is true if the phrase “U-meager sets” is replaced by “U-meager for every cut U.”

We exhibit a finite list of first-order axioms which may be used to define topological spaces. For most separation axioms we discover a first-order equivalent statement.

A countable, atomically stable structure in a finite, relational language has fewer than 2ω non-isomorphic substructures if and only if is cellular. An example shows that the finiteness of the language is necessary.

Let κ be an uncountable cardinal and the edges of a complete graph with κ vertices be colored with ℵ0 colors. For the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if , then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when the edges of the complete graph on vertices are colored with ℵ0 colors. In particular, we are concerned with the case ℵ1 ≤ κ ≤ ℵω.

The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models [2]. When the graph being colored has size ℵ1, the identities that must occur have been classified by Shelah [4]. If the graph has size greater than or equal to ℵω the identities have also been classified in [3]. The number of colors is fixed at ℵ0 as it is the natural place to start and the results here can be generalized to situations where more colors are used.

There is one difference that we now make explicit. When countably many colors are used we can define the following coloring of the complete graph on vertices. First consider the branches in the complete binary tree of height ω to be vertices of a complete graph.

Let

We shall show that every formula in the elementary arithmetic of positive integers may be constructed out of matrices of the form ϕ(x, y) by means of quantifiers and the stroke. For Julia Robinson has shown that the successor function and the relation of divisibility are sufficient to define addition and multiplication; but from elementary considerations

and

Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number of maximal sets whose many-one degrees are pairwise incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′.

The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x ∈ , where M is a maximal set, and M is its complement. We showed that / 1 ≡ / 2 if, and only if, 1 = m 2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree.

It is probably because intuitionism is founded on the concept of (abstract) proof that it has been possible to develop various kinds of models. The following is but a partial list: Gabbay [5], Beth [2], Kripke [8], Kleene [7], Läuchli [9], McKinsey and Tarski [10], Rasiowa and Sikorski [14], Scott [15], de Swart [16], and Veldman [17].

The original purpose for having the models appears to have been for obtaining independence or consistency results for certain formalizations of intuitionism [see Beth [2], Prawitz [13]]; of course, if the models could be also justified as being plausible interpretations of intuitionistic thinking, so much the better. In fact, having some kind of plausible interpretation makes it much easier to work with the models. Occasionally the models were used to suggest possible extensions of the formal systems; for example, the Kripke models with constant domains have motivated interest in the formal logic CD which extends the Intuitionistic Predicate Calculus (IPC) by having the axiom schema

We give a necessary and sufficient condition in order that a type-shifting automorphism be constructed on a model of the Theory of Simple Types (TST) by forcing. Namely it is proved that, if for every n ≥ 1 there is a model of TST in the ground model M of ZFC that contains an n-extendible coherent pair, then there is a generic extension M[G] of M that contains a model of TST with a type-shifting automorphism, and hence M[G] contains a model of NF. The converse holds trivially. It is also proved that there exist models of TST containing 1-extendible coherent pairs.

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):

(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.

(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.

(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.

In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).