On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

A note on canonical functions

We construct a generic extension in which the ℵ₂nd canonical function on ℵ₁ exists. Supported by NSF and by a Fulbright grant. Publ. 378. Partially supported by the B.S.F.     

A simple technique for surface labeling of solids with222Rn and its short-lived decay products

A simple radioactive labeling technique is described which is based on a recoil injection of atoms of²²²Rn and its decay products from a²²⁶Ra source into thin surface layers of solids.     

A Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

Simple complete Boolean algebras

For every regular cardinal there exists a simple complete Boolean algebra with generators.

     

On reflection of stationary sets in

Let be an inaccessible cardinal, and let and is regular and . It is consistent that the set is stationary and that every stationary subset of reflects at almost every .

     

Finite Left-Distributive Algebras and Embedding Algebras

We consider algebras with one binary operation · and one generator (monogenic) and satisfying the left distributive lawa·(b·c)=(a·b)·(a·c). One can define a sequence of finite left-distributive algebrasA_n, and then take a limit to get an infinite monogenic left-distributive algebraA_∞. Results of Laver and Steel assuming a strong large cardinal axiom imply thatA_∞is free; it is open whether the freeness ofA_∞can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain algebra of increasing functions on natural numbers, called anembedding algebra. Using this and results of the first author, we conclude that the freeness ofA_∞is unprovable in primitive recursive arithmetic.     

Complete CCC Boolean Algebras, the Order Sequential Topology, and a Problem of Von Neumann

Let B be a complete ccc Boolean algebra and let _s be the topologyon B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonemptyopen set in (B, _s) is topologically dense.
2. It is consistentthat every weakly distributive complete cccBoolean algebracarries a strictly positive Maharam submeasure.
3. The topologicalspace(B, _s) is sequentially compact if and onlyif the genericextension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but notindependent reals. 2000 Mathematics Subject Classification 28A60,06E10 (primary); 03E55, 54A20, 54A25 (secondary).     

ω 1 can be measurable

It is shown that ifZF + the axiom of choice + “there is a measurable cardinal” is consistent thenZF + “ω ₁ is measurable” is consistent. The corresponding model is a symmetric submodel of the Cohen-type extension which collapses the first measurable cardinal onto ω₀.     

Projective Stationary Sets and a Strong Reflection Principle

The paper studies projective stationary sets. The ProjectiveStationary Reflection Principle is the statement that everyprojective stationary set contains an increasing continuous-chain of length ₁. It is shown that, if Martin's Maximum holds,then the Projective Stationary Reflection Principle holds. Also,this principle is equivalent to the Strong Reflection Principle.The paper shows that the saturation of the nonstationary idealon ₁ is equivalent to a certain kind of reflection.