We give an application of the extender based Radin forcing to cardinal arithmetic. Assuming $\kappa$ is a large enough cardinal we construct a model satisfying $2^{\kappa} = \kappa^{+n}$ together with $2^{\lambda} = \lambda^{+n}$ for each cardinal $\lambda < \kappa$, where $0 < n < \omega$. The cofinality of $\kappa$ can be set arbitrarily or $\kappa$ can remain inaccessible. When $\kappa$ remains an inaccessible, $V_{\kappa}$ is a model of ZFC satisfying $2^{\lambda} = \lambda^{+n}$ for all cardinals $\kappa$.

The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found.

We show that each of Δ¹₃-CA₀ + Σ¹₃-IND and Π¹₂-CA₀ + Π¹₃-TI proves Δ⁰₃-Det and that neither Σ₃¹-IND nor Π¹₃-TI can be dropped. We also show that neither Δ¹₃-CA₀ + Σ¹_∞-IND nor Π¹₂-CA₀ + Π¹_∞-TI proves Σ⁰₃-Det. Moreover, we prove that none of Δ₂¹-CA₀, Σ₃¹-IND and Π₂¹-TI is provable in Δ¹₁-Det₀ = ACA₀ + Δ¹₁-Det.

The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the collection of inductively generated sublocales has coframe structure. Overt sublocales and weakly closed sublocales are described, and related via a new notion of “rest closed” sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with “lower powerpoints”, arising from the impredicative theory of the lower powerlocale. Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with “upper powerpoints”, arising from the impredicative theory of the upper powerlocale.

Let $\mathcal{A}_1,..., \mathcal{A}_n, \mathcal{A}_{n+1}$ be a finite sequence of algebras of sets given on a set $X, \cup^n_{k=1}\mathcal{A}_k \neq \mathfrak{B}(X)$, with more than $\frac{4}{3}n$ pairwise disjoint sets not belonging to $\mathcal{A}_n$. It was shown in [4] and [5] that in this case $\cup^{n+1}_{k=1}\mathcal{A}_k \neq \mathfrak{B}(X)$. Let us consider, instead $\mathcal{A}_{n+1}$, a finite sequence of algebras $\mathcal{A}_{n+1},..., \mathcal{A}_{n+1}$. It turns out that if for each natural $i \leq l$ there exist no less than $\frac{4}{3}(n+l)-\frac{l}{24}$ pairwise disjoint sets not belonging to $\mathcal{A}_{n+i}$, then $\cup^{n+1}_{k=1}\mathcal{A}_k \neq \mathfrak{B}(X)$. But if $l \geq$ 195 and if for each natural $i \leq l$ there exist no less than $\frac{4}{3}(n+l)-\frac{l}{15}$ pairwise disjoint sets not belonging to $\mathcal{A}_{n+i}$, then $\cup^{n+1}_{k=1}\mathcal{A}_k \neq \mathfrak{B}(X)$. After consideration of finite sequences of algebras, it is natural to consider countable sequences of algebras. We obtained two essentially important theorems on a countable sequence of almost $\sigma$-algebras (the concept of almost $\sigma$-algebra was introduced in [4]).

We generalize the Bolzano-Weierstrass theorem (that every bounded sequence of reals admits a convergent subsequence) on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.

We observe that the W^*-hierarchy, a variant (introduced by Downey, Fellows, and Taylor [7]) of the better known W-hierarchy, coincides with the W-hierarchy, though not level wise, but just as a whole hierarchy. More precisely, we prove that W[t]⊆W^*[t]⊆ W[2t-2] for each t≥ 2. It was known before that W[1]=W^*[1] and W[2]=W^*[2]. Our second main result is a new logical characterization of the W^*-hierarchy in terms of “Fagin-definable problems.” As a by-product, we also obtain an improvement of our earlier characterization of the hierarchy in terms of model-checking problems. Furthermore, we obtain new complete problems for the classes W[3] and W^*[3].

Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called ℰ-connection approach for combining modal logics to an algebraic setting.

We consider successor-invariant first-order logic (FO + succ)_inv, consisting of sentences Φ involving an “auxiliary” binary relation S such that (𝔞,S₁) ⊨ Φ {⋔} (𝔞,S₂) ⊨ Φ for all finite structures 𝔞 and successor relations S₁,S₂ on 𝔞. A successor-invariant sentence Φ has a well-defined semantics on finite structures 𝔞 with no given successor relation: one simply evaluates Φ on (𝔞,S) for an arbitrary choice of successor relation S. In this article, we prove that (FO + succ)_inv is more expressive on finite structures than first-order logic without a successor relation. This extends similar results for order-invariant logic [8] and epsilon-invariant logic [10].

In [2] John Burgess describes predicative versions of Frege's logic and poses the problem of finding their exact arithmetical strength. I prove here that PV, the simplest such theory, is equivalent to Robinson's arithmetical theory Q.

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

We give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories T$^2_2$ T$^3_2$.

We show, for any ordinal γ ≥ 3, that the class ℜ𝔞CA_γ is pseudo-elementary and has a recursively enumerable elementary theory. S_cK denotes the class of strong subalgebras of members of the class K. We devise games, Fⁿ (3≤ n≤ω), G, H, and show, for an atomic relation algebra 𝒜 with countably many atoms, that

• ∃ has a winning strategy in F^ω(At(𝒜))⋔ 𝒜∈S_cℜ𝔞CA_ω,

• ∃ has a winning strategy in Fⁿ(At(𝒜)) ⇐ 𝒜∈S_cℜ𝔞CA_n,

• ∃ has a winning strategy in G(At(𝒜)) ⇐ 𝒜∈ℜ𝔞CA_ω,

• ∃ has a winning strategy in H(At(𝒜))→𝒜∈ℜ𝔞RCA_ω

ℜ𝔞RCA_γ ⊆ K ⊆ S_cℜ𝔞CA₅,

that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion ℜ𝔞CA_γ⊂S_cℜ𝔞CA_γ is strict. For infinite γ and for a countable relation algebra 𝒜 we show that 𝒜 has a complete representation if and only if 𝒜 is atomic and ∃ has a winning strategy in F(At(𝒜)) if and only if 𝒜 is atomic and 𝒜∈S_cℜ𝔞CA_γ.

In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ₁-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ₁-elementarity as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.