Let $${\fancyscript{L}}$$ be a finite first-order language and $${\langle{\fancyscript{M}_n} \,|\, {n < \omega}\rangle}$$ be a sequence of finite $${\fancyscript{L}}$$ -models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ^ω2 is non-empty, then there is a non-principal ultrafilter $${\fancyscript{U}}$$ over ω such that the corresponding ultraproduct $${\prod_\fancyscript{U}\fancyscript{M}_n}$$ has an automorphism that is not induced by an element of $${\prod_{n<\omega}{\rm Aut}(\fancyscript{M}_n)}$$ .

Abstract.

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the meager ideal was established by Bartoszyński and Kada.

We study the Lindenbaum algebra $${\fancyscript{A}}$$ (WKL_o, RCA_o) of sentences in the language of second-order arithmetic that imply RCA_o and are provable from WKL_o. We explore the relationship between $${\Sigma^1_1}$$ sentences in $${\fancyscript{A}}$$ (WKL_o, RCA_o) and $${\Pi^0_1}$$ classes of subsets of ω. By applying a result of Binns and Simpson (Arch. Math. Logic 43(3), 399–414, 2004) about $${\Pi^0_1}$$ classes, we give a specific embedding of the free distributive lattice with countably many generators into $${\fancyscript{A}}$$ (WKL_o, RCA_o).

Abstract.

 The paper establishes lower bounds on the provability of 𝒟=NP and the MRDP theorem in weak fragments of arithmetic. The theory I⁵E₁ is shown to be unable to prove 𝒟=NP. This non-provability result is used to show that I⁵E₁ cannot prove the MRDP theorem. On the other hand it is shown that I¹E₁ proves 𝒟 contains all predicates of the form (∀i≤|b|)P(i,x)^Q(i,x) where ^ is =, <, or ≤, and I⁰E₁ proves 𝒟 contains all predicates of the form (∀i≤b)P(i,x)=Q(i,x). Here P and Q are polynomials. A conjecture is made that 𝒟 contains NLOGTIME. However, it is shown that this conjecture would not be sufficient to imply 𝒟=N P. Weak reductions to equality are then considered as a way of showing 𝒟=NP. It is shown that the bit-wise less than predicate, ≤₂, and equality are both co-NLOGTIME complete under FDLOGTIME reductions. This is used to show that if the FDLOGTIME functions are definable in 𝒟 then 𝒟=N P.

Abstract.

Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary first-order predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PL-proofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the β-normal-form of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended language, the terms are ordinary PL-terms. In contrast to approaches to Herbrand’s theorem based on cut elimination orɛ-elimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrand-term extraction tool.

We define a theory of two-sort bounded arithmetic whose provably total functions are exactly those in $${\mathcal{F}_{LOGCFL}}$$ by way of a generalized quantifier that expresses computations of SAC¹ circuits. The proof depends on Kolokolova’s conditions for the connection between the provable capture in two-sort theories and descriptive complexity.

Abstract.

 In this paper we study and equationally characterize the subvarieties of BL, the variety of BL-algebras, which are generated by families of single-component BL-chains, i.e. MV-chains, Product-chain or Gödel-chains. Moreover, it is proved that they form a segment of the lattice of subvarieties of BL which is bounded by the Boolean variety and the variety generated by all single-component chains, called ŁΠG.

We extend the construction of a global square sequence in extender models from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s characterization in L.

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of $$2^\kappa $$ , $$\kappa $$ inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah (On CON(Dominating $$\_$$ lambda $$\,>\,$$ cov $$\_\lambda $$ (meagre)), arXiv:0904.0817 , Problem 0.5) and concerns the definition of a forcing which is $$\kappa ^\kappa $$ -bounding, $$<\kappa $$ -closed and $$\kappa ^+$$ -cc, for $$\kappa $$ inaccessible. Cohen and Shelah (Generalizing random real forcing for inaccessible cardinals, arXiv:1603.08362 ) provide a proof for (Shelah, On CON(Dominating $$\_$$ lambda $$\,>\,$$ cov $$\_\lambda $$ (meagre)), arXiv:0904.0817 , Problem 0.5), and in this paper we independently reprove this result by using a different type of construction. This also contributes to a line of research adressed in the survey paper (Khomskii et al. in Math L Q 62(4–5):439–456, 2016).