α-modules and generalized submodules

A QTAG-module M is an α-module, where α is a limit ordinal, if M/Hβ (M) is totally projective for every ordinal β < α. In the present paper α-modules are studied with the help of α-pure submodules, α-basic submodules, and α-large submodules. It is found that an α-closed α-module is an α-injective. For any ordinal ω ≤ α ≤ ω1 we prove that an α-large submodule L of an ω1-module M is summable if and only if M is summable.

A direct proof of a theorem of Jech and Shelah on PCF algebras

By using an argument based on the structure of the locally compact scattered spaces, we prove in a direct way the following result shown by Jech and Shelah: there is a family {Bα : α < ω1} of subsets of ω1 such that the following conditions are satisfied: (a) max Bα - α, (b) if α ∈ Bβ then Bα ⊆ Bβ, (c) if δ ≤ α and δ is a limit ordinal then Bα ∩ δ is not in the ideal generated by the sets Bβ, β < α, and by the bounded subsets of δ, (d) there is a partition {An : n ∈ ω} of ω1 such that for every α and every n, Bα ∩An is finite.

ISOMORPHISM ON HYP

We show that isomorphism is not a complete ${\rm{\Sigma }}_1^1$ equivalence relation even when restricted to the hyperarithmetic reals: If E1 denotes the ${\rm{\Sigma }}_1^1$ (even ${\rm{\Delta }}_1^1$) equivalence relation of [4] then for no Hyp function f do we have xEy iff f(x) is isomorphic to f(y) for all Hyp reals x,y. As a corollary to the proof we provide for each computable limit ordinal α a hyperarithmetic reduction of ${ \equiv _\alpha }$ (elementary-equivalence for sentences of quantifier-rank less than α) on arbitrary countable structures to isomorphism on countable structures of Scott rank at most α.

Intrinsic bounds on complexity and definability at limit levels

We show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

ELEMENTARY AMENABLE SUBGROUPS OF R. THOMPSON'S GROUP F

We study the subgroups of R. J. Thompson's group F and PLo(I), the group of orientation preserving, piecewise linear self homeomorphisms of [0, 1]. We exhibit, for each non-limit ordinal α ≤ ω2 + 1, an elementary amenable group of elementary class α (under Chou's stratification of elementary amenable groups) that is a subgroup of F and thus of PLo(I). We also give examples that negatively answer a question of Sapir about non-solvable groups in F and PLo(I).

Slim models of Zermelo set theory

Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence ⟨Aλ ∣ λ a limit ordinal⟩ where for each λ. Aλ ⊆ λ2, there is a supertransitive inner model of Zermelo containing all ordinals in which for every λAλ = {a ∣ Φ(λ, a)}.

Possible PCF algebras

There exists a family of sets of countable ordinals such that:(1) max Bα = α,(2) if α ∈ Bβ then Bα ⊆ Bβ,(3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ, β < α, and by the bounded subsets of λ,(4) there is a partition of ω1 such that for every α and every n, Bα ∩ An is finite.

Building iteration trees

It is shown, assuming the existence of a Woodin cardinal δ, that every tree ordering on some limit ordinal λ < δ with a cofinal branch is the tree ordering of some iteration tree on V.

On the weak Kleene scheme in Kripke's theory of truth

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered:1. There exist sentences that are neither paradoxical nor grounded.2. There are fixed points.3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n ∣ A(n) is true in the minimal fixed point}, where A(x) is a formula of AR + T) are precisely the sets.4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the sets.5. The closure ordinal for Kripke's construction of the minimal fixed point is .In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen:1. There may or may not exist nonparadoxical, ungrounded sentences.2. The number of fixed points may be any positive finite number, ℵ0, or .3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the sets, including intermediate cases.4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the sets.5. The closure ordinal for the construction of the minimal fixed point may be ω, , or any successor limit ordinal in between.In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Gödel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language.

The ordertype of β-R.E. sets

Let β be an arbitrary limit ordinal. A β-r.e. set is l-finite iff all its β-r.e. subsets are β-recursive. The l-finite sets correspond to the ideal of finite sets in the lattice of r.e. sets. We give a characterization of l-finite sets in terms of their ordertype: a β-r.e. set is l-finite iff it has ordertype less than β*, the Σ1, projectum of β).