Mathematics

Given any 1-random set X and any r∈(0,1) , we construct a set of intrinsic density r which is computable from r⊕X . For almost all r , this set will be the first known example of an intrinsic density r set which cannot compute any r -Bernoulli random set. To achieve this, we shall formalize the {\tt into} and {\tt within} noncomputable coding methods which work well with intrinsic density.

We continue the work done by Gitik, Kanovei, Koepke, and later by the authors. We prove that for every set A in a Magidor-Radin generic extension using a coherent sequence such that o U → (κ)<κ , there is a subset C ′ of the Magidor club such that V[A]=V[ C ′ ] . Also we classify all intermediate ZFC transitive models V⊆M⊆V[G] .

In this paper from 2009 we study IL(PRA), the interpretability logic of PRA. As PRA is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to PRA: IL(PRA) is not ILM or ILP. IL(PRA) does of course contain all the principles known to be part of IL(All), the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of PRA and see what their consequences in the modal logic IL(PRA) are. These properties are reflected in the so-called Beklemishev Principle B, and Zambella's Principle Z, neither of which is a part of IL(All). Both principles and their interrelation are submitted to a modal study. In particular, we prove a frame condition for B. Moreover, we prove that Z follows from a restricted form of B. Finally, we give an overview of the known relationships of IL(PRA) to important other interpetability principles.

Let M=?�K;O??be a real closed valued field and let k be its residue field. We prove that every interpretable field in M is definably isomorphic to either K , K( ?? ????????) , k , or k( ?? ????????) . The same result holds when K is a model of T , for T an o-minimal power bounded expansion of a real closed field, and O is a T -convex subring. The proof is direct and does not make use of known results about elimination of imaginaries in valued fields.

We improve on and generalize a 1960 result of Maltsev. For a field F , we denote by H(F) the Heisenberg group with entries in F . Maltsev showed that there is a copy of F defined in H(F) , using existential formulas with an arbitrary non-commuting pair (u,v) as parameters. We show that F is interpreted in H(F) using computable Σ 1 formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in H(F) of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in H(F) . Looking at what was used to arrive at this parameter-free interpretation of F in H(F) , we give general conditions sufficient to eliminate parameters from interpretations.

Consider two paths ϕ,ψ:[0;1]→[0;1 ] 2 in the unit square such that ϕ(0)=(0,0) , ϕ(1)=(1,1) , ψ(0)=(0,1) and ψ(1)=(1,0) . By continuity of ϕ and ψ there is a point of intersection. We prove that from ϕ and ψ we can compute closed intervals S ϕ , S ψ ⊆[0;1] such that ϕ( S ϕ )=ψ( S ψ ) .

The paper is an introduction to intuitionistic mathematics.

The first seeds of mathematical intuitionism germinated in Europe over a century ago in the constructive tendencies of Borel, Baire, Lebesque, Poincaré, Kronecker and others. The flowering was the work of one man, Luitzen Egbertus Jan Brouwer, who taught mathematics at the University of Amsterdam from 1909 until 1951. By proving powerful theorems on topological invariants and fixed points of continuous mappings, Brouwer quickly build a mathematical reputation strong enough to support his revolutionary ideas about the nature of mathematical activity. These ideas influenced Hilbert and Gödel and established intuitionistic logic and mathematics as subjects worthy of independent study. Our aim is to describe the development of Brouwer's intuitionism, from his rejection of the classical law of excluded middle to his controversial theory of the continuum, with fundamental consequences for logic and mathematics. We borrow Kleene's formal axiomatic systems (incorporating earlier attempts by Kolmogorov, Glivenko, Heyting and Peano) for intuitionistic logic and arithmetic as subtheories of the corresponding classical theories, and sketch his use of gödel numbers of recursive functions to realize sentences of intuitionistic arithmetic including a form of Church's Thesis. Finally, we present Kleene and Vesley's axiomatic treatment of Brouwer's continuum, with the function-realizability interpretation which establishes its consistency.

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ω ω -bounding forcing notions, 2) the class of subproper, T -preserving forcing notions (where T is a fixed Souslin tree) and 3) the class of subproper, [T] -preserving forcing notions (where T is an ω 1 -tree) are iterable with revised countable support. In the second part, we adopt Miyamoto's theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve ω 1 , and, in the case of subcompleteness, don't add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

In 'Some Remarks on Extending an Interpreting Theories with a Partial Truth Predicate' Reinhardt famously proposed an instrumentalist interpretation of the truth theory Kripke-Feferman (KF) in analogy to Hilbert's program. Reinhardt suggested to view KF as a tool for generating 'the significant part of KF', that is, as a tool for deriving sentences of the form T\ulcorner\varphi\urcorner . The constitutive question of Reinhardt's program was whether it was possible "to justify the use of nonsignificant sentences entirely within the framework of significant sentences"? This question was answered negatively by Halbach and Horsten (2006) but we argue that under a more careful interpretation the question may receive a positive answer. To this end, we propose to shift attention from KF-provably true sentences to KF-provably true inferences, that is, we shall identify the significant part of KF with the set of pairs \langle\Gamma, \Delta\rangle , such that KF proves that if all members of \Gamma are true, at least one member of \Delta is true. In way of addressing Reinhardt's question we show that the provably true inferences of KF coincide with the provable sequents of the theory Partial Kripke-Feferman (PKF).