Countable OD sets of reals belong to the ground model

It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \(\text {OD}\) elements.     

Some natural equivalence relations in the Solovay model

We obtain some non-reducibility results concerning some natural equivalence relations on reals in the Solovay model. The proofs use the existence of reals x which are minimal with respect to the cardinals in L[x], in a certain sense. S.-D. Friedman research was supported by Grant P 19375-N18 of the Austrian Science Fund (FWF). V. Kanovei research was supported by Grants 06-01-00608 and 07-01-00445 of the Russian Foundation for Basic Research (RFBR).     

On Shattering,Splitting and Reaping Partitions

In this article we investigate the dual-shattering cardinal ?, the dual-splitting cardinal ?? and the dual-reaping cardinal ??, which are dualizations of the well-known cardinals ?? (the shattering cardinal, also known as the distributivity number of P(ω)/fin), s (the splitting number) and ?? (the reaping number). Using some properties of the ideal ?? of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of ω, we show that add(??) = cov(??) = ?. With this result we can show that ? > ω₁ is consistent with ZFC and as a corollary we get the relative consistency of ? > ?? t, where t is the tower number. Concerning ?? we show that cov(M) ? ?? ?? (where M is the ideal of the meager sets). For the dual-reaping cardinal ?? we get p ?? ? ?? ? ?? (where ?? is the pseudo-intersection number) and for a modified dual-reaping number ??′ we get ??′ ? ?? (where ?? is the dominating number). As a consistency result we get ?? < cov(??).     

Selective ultrafilters and homogeneity

We develop a game-theoretic approach to partition theorems, like those of Mathias, Taylor, and Louveau, involving ultrafilters. Using this approach, we extend these theorems to contexts involving several ultrafilters. We also develop an analog of Mathias forcing for such contexts and use it to show that the proposition (considered by Laver and Prikry) “every non-trivial c.c.c. forcing adjoins Cohen-generic reals or random reals” implies the non-existence of P-points. We show that, in the model obtained by Lévy collapsing to ω all cardinals below a Mahlo cardinal ;, any countably many selective ultrafilters are mutually generic over the Solovay (Lebesgue measure) submodel. Finally, we show that a certain natural group of self-homeomorphisms of βω-ω, chosen so as to preserve selectivity of ultrafilters, in fact preserves isomorphism types.     

Hereditary properties of the class of closed sets of uniqueness for trigonometric series

It is shown that theσ-idealU ₀ of closed sets of extended uniqueness inT is hereditarily non-Borel, i.e. every “non-trivial”σ-ideal of closed setsI⊆U ₀ is non-Borel. This implies both the result of Solovay, Kaufman that bothU ₀ andU (theσ-ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset ofT of extended uniqueness is of the first category. A further extension to ideals contained inU ₀ is given. Research partially supported by NSF Grant DMS-8718847.     

Resolution of the uniform lower bound problem in constructive analysis

In a previous paper we constructed a full and faithful functor ?? from the category of locally compact metric spaces to the category of formal topologies (representations of locales). Here we show that for a real‐valued continuous function f, ??(f) factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than point‐wise positivity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Right invariant right holoids

Right invariant right holoids H (r.i.r. holoids) are totally (and positively) ordered semigroups such that a > b holds if and only if a = bc for some c ¦ e e the identity of H. These holoids occur as semigroups of the prinicipal right ideals of right invariant right chain rings. We investigate in which way r.i.r. holoids of finite rank are built up from r.i.r. holoids of rank one which are known to be subsemigroups of the non-negative real numbers under addition. This is best described by conditions on f(C,a) where C is a prime segment which is shifted over elements a ? H. The functional properties of f(C,a) are studied, especially in the finite-rank-case. These results are then applied to the extension problem. Here, conditions are given under which the extension splits, however even under these assumptions aft additional problem occurs. An element s is called a denominator for b if a solution x exists in H with xs — b. It is crucial to know the denominators sets and solution sets. Under certain condition it is possible to embed H into a r.i.r. holoid H′ with larger sets of denominators.     

On a convenient property about $${[\gamma]^{\aleph_0}}$$

There is a partial order \mathbbP{\mathbb{P}} preserving stationary subsets of ω ₁ and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω ₁ over V also collapses ω ₁ over V^\mathbbP{V^{\mathbb{P}}} . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B.     

Proper and piecewise proper families of reals

I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ ω₁. Here, I investigate the question of the existence of proper and piecewise proper families of reals of different cardinalities. I show that it is consistent relative to ZFC to have continuum many proper families of cardinality ω₁ and continuum many piecewise proper families of cardinality ω₂ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ulm Classification of Analytic Equivalence Relations in Generic Universes

We prove that if every real belongs to a set generic extension of L, then every Σ equivalence relation E on reals either admits a Δ₁ reduction to the equality on the set 2^{< ω1} of all countable binary sequences, or the Vitali equivalence E₀ continuously embeds in E. The proofs are based on a topology generated by OD sets.     

Fermat–Reyes method in the ring of Fermat reals

To discover derivatives, Pierre de Fermat used to assume a non-zero increment h in the incremental ratio and, after some calculations, to set h=0 in the final result. This method, which sounds as inconsistent, can be perfectly formalized with the Fermat–Reyes theorem about existence and uniqueness of a smooth incremental ratio. In the present work, we will introduce the cartesian closed category where to study and prove this theorem and describe in general the Fermat method. The framework is the theory of Fermat reals, an extension of the real field containing nilpotent infinitesimals which does not need any knowledge of mathematical logic. This key theorem will be essential in the development of differential and integral calculus for smooth functions defined on the ring of Fermat reals and also for infinite-dimensional operators like derivatives and integrals.     

On the separation of regularity properties of the reals

We present a model where ω ₁ is inaccessible by reals, Silver measurability holds for all sets but Miller and Lebesgue measurability fail for some sets. This contributes to a line of research started by Shelah in the 1980s and more recently continued by Schrittesser and Friedman (see [7]), regarding the separation of different notions of regularity properties of the real line.     

The computable Lipschitz degrees of computably enumerable sets are not dense

The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truth-table reducibility (Downey et al. (2004) [6]). This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’ proof that the identity bounded Turing degrees of c.e. sets are not dense (George Barmpalias, Andrew E.M. Lewis (2006) [2]), however the problem for the computable Lipschitz degrees is more complex.     

Products of operator ideals and extensions of Schatten classes

We study relations between Schatten classes and product operator ideals, where one of the factors is the Banach ideal Π_E,2 of (E, 2)‐summing operators, and where E is a Banach sequence space with ?₂ ? E. We show that for a large class of 2‐convex symmetric Banach sequence spaces the product ideal Π_E,2 ○ ??^a_q,s is an extension of the Schatten class ??_F with a suitable Lorentz space F. As an application, we obtain that if 2 ≤ p, q < ∞, 1/r = 1/p + 1/q and E is a 2‐convex symmetric space with fundamental function λ_E(n) ≈? n^1/p, then Π_E,2 ○ Π_q is an extension of the Schatten class ??r,q (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Definable Elements of Definable Borel Sets

Mathematical Notes - We prove that it is true in Sacks, Cohen, and Solovay generic extensions that any ordinal definable Borel set of reals necessarily contains an ordinal definable element. This...     

Projective Well-orderings of the Reals

If there is no inner model with ω many strong cardinals, then there is a set forcing extension of the universe with a projective well-ordering of the reals.     

Arithmetic progressions in sumsets

We prove several results concerning arithmetic progressions in sets of integers. Suppose, for example, that a \alpha and b \beta are positive reals, that N is a large prime and that C,D í \Bbb Z/N\Bbb Z C,D \subseteq {\Bbb Z}/N{\Bbb Z} have sizes gN \gamma N and dN \delta N respectively. Then the sumset C + D contains an AP of length at least e^{c ?{log} N} e^{c \sqrt{\rm log} N} , where c > 0 depends only on g \gamma and d \delta . In deriving these results we introduce the concept of hereditary non-uniformity (HNU) for subsets of \Bbb Z/N\Bbb Z {\Bbb Z}/N{\Bbb Z} , and prove a structural result for sets with this property.     

Undecidability results for restricted universally quantified formulae of set theory

The problem of establishing whether there are sets satisfying a formula in the first order set theoretic language ??_? based on =,?, which involves only restricted quantifiers and has an equivalent ??-prenex form ((??)₀-formula), is neither decidable nor semidecidable. In fact, given any ω-model of ZF – FA, where FA denotes the Foundation Axiom, the set of existential closures of (??)₀-formulae true in the model is a productive set. Undecidability arises even when dealing with restricted universal quantifiers only, provided a predicate is_a_pair(x), meaning that x is a pair of distinct sets, is added to ??_?. If satisfiability refers to ω-models of ZF – FA in which a form of Boffa's antifoundation axiom holds, then semidecidability fails as well; in fact, given any such model, the set of existential closures of formulae involving only restricted quantifiers and the predicate is_a_pair which are true in it, is a productive set. These results are all proved by making use of appropriate codings of Turing machine computations in the set theoretic language.     

Substructure lattices and almost minimal end extensions of models of Peano arithmetic

This paper concerns intermediate structure lattices Lt(??/??), where ?? is an almost minimal elementary end extension of the model ?? of Peano Arithmetic. For the purposes of this abstract only, let us say that ?? attains L if L ? Lt(??/??) for some almost minimal elementary end extension of ??. If T is a completion of PA and L is a finite lattice, then: (A) If some model of T attains L, then every countable model of T does. (B) If some rather classless, ?₁‐saturated model of T attains L, then every model of T does. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Binary subtrees with few labeled paths

We prove several quantitative Ramseyan results involving ternary complete trees with {0,1}-labeled edges where we attempt to find a complete binary subtree with as few labels as possible along its paths. One of these is used to answer a question of Simpson??s in computability theory; we show that there is a bounded ?? ₁ ⁰ class of positive measure which is not strongly (Medvedev) reducible to DNR₃; in fact, the class of 1-random reals is not strongly reducible to DNR₃.