On what I do not understand (and have something to say): Part I

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (``see ... '' means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear

Embedding Theorem for the automorphism group of the α-enumeration degrees

It is a theorem of classical Computability Theory that the automorphism group of the enumeration degrees D_e embeds into the automorphism group of the Turing degrees D_T . This follows from the following three statements: 1. D_T embeds to D_e , 2. D_T is an automorphism base for D_e, 3. D_T is definable in D_e . The first statement is trivial. The second statement follows from the Selman’s theorem: A ≤e B ⇐⇒ ∀X ⊆ ω[B ≤e X ⊕ complement(X) implies A ≤e X ⊕ complement(X)]. The third statement follows from the definability of a Kalimullin pair in the α-enumeration degrees D_e and the following theorem: an enumeration degree is total iff it is trivial or a join of a maximal Kalimullin pair. Following an analogous pattern, this thesis aims to generalize the results above to the setting of α-Computability theory. The main result of this thesis is Embedding Theorem: the automorphism group of the α-enumeration degrees D_αe embeds into the automorphism group of the α-degrees D_α if α is an infinite regular cardinal and assuming the axiom of constructibility V = L. If α is a general admissible ordinal, weaker results are proved involving assumptions on the megaregularity. In the proof of the definability of D_α in D_αe a helpful concept of α-rational numbers Q_α emerges as a generalization of the rational numbers Q and an analogue of hyperrationals. This is the most valuable theory development of this thesis with many potentially fruitful directions

Cardinal invariants related to permutation groups

AbstractWe consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers:ag≔ the least cardinal number of maximal cofinitary permutation groups;ap≔ the least cardinal number of maximal almost disjoint permutation families;c(Sym(N))≔ the cofinality of the permutation group on the set of natural numbers.We show that it is consistent with ZFC that ap=ag<c(Sym(N))=ℵ2; in fact we show that in the Miller model ap=ag=ℵ1<ℵ2=c(Sym(N))

Trivial automorphisms

We prove that the statement `For all Borel ideals I and J on $\omega$, every isomorphism between Boolean algebras $P(\omega)/I$ and $P(\omega)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(\omega)/I$ and any other quotient $P(\omega)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $\omega$. We can also assure that the dominating number is equal to $\aleph_1$ and that $2^{\aleph_1}>2^{\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(\omega)/Fin$ are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of suslin proper forcings.Comment: Thoroughly revised versio

On the structure of immediate extensions of valued fields

The connection between a valued eld extension and the corresponding extensions of the value group and the residue eld is meaningful for the theory of valued elds. When this connection is interrupted, the structure of valued eld extensions is much more complicated. This causes one of the main hurdles to solve many important questions in valuation theory and related areas of mathematics. Crucial examples of this situation are defect extensions and immediate extensions of valued elds. A better understanding of both types of extensions turned out to be important for questions in algebraic geometry, like resolution of singularities, problems in real algebra and the model theory of valued elds. In this thesis we study the structure and constructions of immediate as well as defect extensions of valued elds. In particular, we focus on the structure of maximal immediate extensions of valued elds. In connection with local uniformization, a local version of resolution of singularities, we investigate the problems related to defect extensions. We describe properties of distances of elements in valued eld extensions, which turned out to be a useful tool for the study of the structure of defect extensions of valued elds of positive characteristic. We also give an upper bound of the number of distinct distances of immediate elements of a bounded degree. We further study the problem of existence of in nite towers of Galois defect extensions of prime degree. We give conditions for a valued eld to admit such towers and present constructions of them. In connection with questions related to local uniformization we present constructions of in nite towers of Artin-Scheier defect extensions of rational function elds in two variables over elds of positive characteristic. We consider the classi cation of Artin-Schreier defect extensions into \dependent" and \independent" ones (according to whether they are connected with purely inseparable defect extensions, or not). To understand the meaning of the classi cation for the issue of local uniformization, we consider various valuations of the above mentioned rational function elds and investigate for which they admit an in nite tower of Artin-Schreier defect extensions of each type. The existence of in nite towers of Galois defect extensions of prime degree turned out to be important for the structure of maximal immediate extensions of valued elds, which is the next problem treated in this thesis. We give conditions for a valued eld to admit maximal immediate extensions of in nite transcendence degree. This problem is tightly connected with the description of the possible extensions of a valuation from a given eld to an algebraic function eld. We further consider algebraic extensions of maximal elds and study the structure of immediate extensions of such elds. We also investigate the problem of uniqueness of maximal immediate extensions. We prove that there is a class of valued elds which admit an algebraic maximal immediate extension as well as one of in nite transcendence degree, which can be seen as the worst possible case of non-uniqueness. In our studies of maximal immediate extensions we consider also valued elds (K; v) with p-divisible value group and perfect residue eld, where p is the characteristic exponent of the residue eld Kv. Maximal immediate extensions of such elds are tame elds. Because of their good valuation theoretical and model theoretical properties, tame elds play an important role in the theory of valued elds and its applications. We discuss rst the case of elds with maximal immediate extensions of nite transcendence degree and describe the structure of such extensions. We then relate the existence of defect extensions of the eld (K; v) with the structure of the maximal immediate extensions of this eld. We prove that if the eld (K; v) admits a nontrivial separable-algebraic defect extension, then every maximal immediate extension of K is of in nite transcendence degree. We nally apply the results to the description of the structure of valued rational function elds. In particular, we give necessary and su cient conditions on (K; v) to admit an extension of the valuation to a rational function eld F over K such that vF=vK is a torsion group and the residue eld extension FvjKv is algebraic