On a poset algebra which is hereditarily but not canonically well generated

LetB be a superatomic Boolean algebra.B is well generated, if it has a well founded sublatticeL such thatL generatesB. The free product of Boolean algebrasB andC is denoted byB *C. IfC is a chain thenB(C) denotes the interval algebra overC. Theorem 1: (a)Every Boolean subalgebra of B(ℵ₁) *B(ℵ₀)is well-generated. (b)B(ℵ₁) *B(ℵ₁)contains a non well-generated Boolean subalgebra. Canonical well-generatedness is defined in the introduction. Recall thatB(ℵ₁) *B(ℵ₀) is canonically well-generated, and thus well-generated. We prove the following result. Theorem 2:B(ℵ₁) *B(ℵ₀)contains a non canonically well generated Boolean subalgebra. In contrast with Theorem 1(b), we have the following result. Theorem 3:Let A ={ɑ:α<ℵ₁}⊆℘(w)be a strictly increasing sequence in the relation of almost containment. Let B be the subalgebra of ℘(w)generated by {{n}:n∈ℵ₀}∪A.Then B is superatomic, and B is not embeddable in a well-generated algebra.     

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Lifting elementary embeddings <Emphasis Type="Italic">j</Emphasis>: <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript> → <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript>

We describe a fairly general procedure for preserving I₃ embeddings j: V _λ → V _λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I₃ embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in which consistency was established assuming an I₁ embedding.      

Some weak fragments of Martin’s axiom related to the rectangle refining property

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ₁ dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ₁ dense sets related to the rectangle refining property, and prove that they are really weaker fragments. T. Yorioka was partially supported by Grant-in-aids for Scientific Research No.16340022 and No.18840022.     

Determinacy implies that ℵ2 is supercompact

Assuming the axiom of determinacy, ℵ₁ and ℵ₂ areδ-supercompact, whereδ is a fairly large cardinal which will be defined below.     

Multi-dimensionality

We prove that ifT is stable, not multi-dimensional theory, then there is an infinite indiscernible set orthogonal to the empty set. This completes the proof that if ℵ_α=ℵ _•T• ^α >ℵ≥_Kr(T), thenT has ≥2^•α−β• non-isomorphic ℵ_β models of cardinality ℵ_α. Originally written November 5, 1988. Publication 429. Partially supported by the Israel-United States Binational Science Foundation; I thank Alice Leonhardt for the beautiful typing.     

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.     

On the number of homogeneous models of a given power

It is shown that, For each complete theoryT, the nomberh _T(m) of homogeneous models ofT of powerm is a non-increasing function of uncountabel cardinalsm Moreover, ifh _T(ℵ₀)≦ℵ₀, then the functionh _T is also non-increasing ℵ₀ to ℵ₁. This work was supported in part by NSF contracts GP 4257 and GP5913.     

Exponents of someN-compact spaces

For any topological spaceT, S. Mrówka has defined Exp (T) to be the smallest cardinal κ (if any such cardinals exist) such thatT can be embedded as a closed subset of the productN ^κ of κ copies ofN (the discrete space of cardinality ℵ₀). We prove that forQ, the space of the rationals with the inherited topology, Exp (Q) is equal to a certain covering number, and we show that by modifying some earlier work of ours it can be seen that it is consistent with the usual axioms of set theory including the choice that this number equal any uncountable regular cardinal less than or equal to 2^ℵ ₀. Mrówka has also defined and studied the class ℳ={κ: Exp (N _κ)=κ} whereN _κ is the discrete space of cardinality κ. It is known that the first cardinal not in ℳ must not only be inaccessible but cannot even belong to any of the first ω Mahlo classes. However, it is not known whether every cardinal below 2^ℵ ₀ is contained in ℳ. We prove that if there exists a maximal family of almost-disjoint subsets ofN of cardinality κ, then κ∈ℳ, and we then use earlier work to prove that if it is consistent that there exist cardinals which are not in the first ω Mahlo classes, then it is consistent that there exist such cardinals below 2^ℵ ₀ and that ℳ nevertheless contain all cardinals no greater than 2^ℵ ₀. Finally, we consider the relationship between ℳ and certain “large cardinals”, and we prove, for example, that if μ is any normal measure on a measurable cardinal, then μ(ℳ)=0.     

A hierarchy of axiom systemZF

In this paper we develop a sequenceZ ₀, ...,Z _α,... of axiom systems for set theory, such that (1) the consistency of any system within the sequence is provable in its succeeding systems, (2) the first system in the sequence is Zermelo's system Z and the union of all systems in the sequence is justZF. And we prove that for ordinal number α>1, there exists a sequence of ℵ_a+1 axiom systems between systemsZ _α andZ _α+1 such that these systems satisfy the above condition (1).     

Classi localmente filtrali

Riassunto Si dà la definizione di classe ?localmente filtrale?. Si diceche K è una classe localmente filtrale se per ogni n∈ω, per ogni A ₀,...,A _n−1, εK e per ogni famiglia di sotto-insiemi V_i di A_i (i∈n) con V_i finiti, la classe {B ₀,...,B _n−1 delle algebre generate da V₀, ..., V_n−1 è costituita da algebre finite ed è filtrale. Si dimostra che seK è localmente filtrale alloraV _L(K)=IR _L DS(K) e si dà un teorema di caratterizzazione per queste classi.

---

Summary We define a ?classe localmente filtrale? as follows: LetK be a class of similar algebras;K is a ?classe localmente filtrale? if for andn ∈ ω and for anyA ₀,...,A _n−1 ink and for any family of finite subsetsV _i ofA _i(i∈n), the class {B ₀,...,B _n−1 of algebras generated byV _o, ...,V _n−1 consists of finite algebras and is ?filtrale?. We show that ifK is ?localmente filtrale? thenV _L (K)=IR _L DS(K) and we give a characterization theorem for these classes.

---

---

Lavoro eseguito nell'ambito dei gruppi di ricerca matematici del C.N.R. per l'anno 1970–71.     

On partitions of the real line

Answering a question of Sierpinski, we prove that the real line is not necessarily the disjoint union of {btℵ} ₁ non-emptyG _σ sets.     

Generalized Martin’s Axiom and Souslin’s hypothesis for higher cardinals

We consider different generalizations of Martin’s Axiom to higher cardinals. For ℵ₁, assuming CH+2^ℵ ₁>ℵ₂+□_ℵ1 we show that a generalized Martin’s Axiom considered by Baumgartner settles the ℵ₂ Souslin Hypothesis ... the wrong way. We further show that, assuming CH+2^ℵ ₁>ℵ₂, a strengthening of this axiom implies □_{ℵ 1}. Finally, we show that a seemingly innocuous further strengthening is inconsistent with CH+2^ℵ ₁>ℵ₂. This author thanks the US-Israel Binational Science Foundation for partial support of this research.     

Reduced products which are not saturated

Summary IfT is a complete theory of Boolean algebra, then we writeA ⊲_T B to denote that for every cardinal κ and every κ-regular filter over a setI such that the Boolean algebra 2 _F ^I of all subsets ofI reduced byF is a model ofT, the reduced powerA _F ^I isK ⁺-saturated wheneverB _F ^I isK ⁺-saturated. The relation ⊲_T generalizes the relation ◃ introduced by Keisler. As in the case of Keisler's ◃ it happens that ⊲_T’s are relations between complete theories, i.e. ifA≡B thenA ⊲_T B andB ⊲_T A. In this paper some examples of theories which are maximal (minimal) with respect to ⊲_T’s are provided and the relations ⊲_T are compared with each other. Presented by J. Mycielski     

Approximation of analytic by Borel sets and definable countable chain conditions

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ ₁ ¹ countable chain condition if there is aΣ ₁ ¹ equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ ₁ ¹ set is equal to a Borel set modulo a set fromI iffI fulfils theΣ ₁ ¹ countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ ₁ ¹ sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition. Research partially supported by NSF grant DMS-9317509.     

Some results on consecutive large cardinals II: Applications of radin forcing

Letκ be a 3 huge cardinal in a countable modelV of ZFC, and letA andB be subsets of the successor ordinals <κ so thatA ⋃B={α<κ:α is a successor ordinal}. Using techniques of Gitik, we construct a choiceless modelN _A of ZF of heightκ so thatN _A ╞“ZF+⌍AC _ω+Forα ∈A, ℵ_a is a Ramsey cardinal+Forβ ∈B, ℵ_β is a singular Rowbottom cardinal which carries a Rowbottom filter+Forγ a limit ordinal, ℵ_y is a Jonsson cardinal which carries a Jonsson filter”. The author wishes to express his thanks to the Rutgers Research Council for a Summer Research Fellowship which partially supported this work. The author also wishes to thank Moti Gitik and Bob Mignone for their useful comments concerning the subject matter of this paper.     

On embeddings of some quotient algebras of free sums of Lie algebras

Let (B _i ) _i∈I be a set of Lie algebras; let X be a free Lie algebra; let * X be their free sum; let R be an ideal of F such that R ⋂ B _i = 1 (i ∈ I); let V be a variety of Lie algebras such that V(R) is an ideal of F. Under some restrictions, we construct an embedding of F/V(R) into the verbal wreath product of a free algebra of the variety V with F/R. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 235–241, 2004.     

The number of non-isomorphic models of an unstable first-order theory

It is proved that ifT is an unstable (first-order) theory,λ>|T|+ℵ₀, thenT has exactly 2 ^λ non-isomorphic models of cardinalityλ. In fact we have stronger results: this is true for pseudo-elementary classes, and for almost everyλ≧|T|+ℵ₁. The preparation of this paper was sponsored in part by NSF Grant GP-22937. This work was supported in part by NSF Grant GP-22794.     

Violating the singular cardinals hypothesis without large cardinals

We extend a transitive model V of ZFC+GCH cardinal preservingly to a model N of ZF + “GCH holds below ℵ _ω ” + “there is a surjection from the power set of ℵ _ω onto λ”, where λ is an arbitrarily high fixed cardinal in V. The construction can be described as follows: add ℵ _n +1 many Cohen subsets of ℵ _n+1 for every n < ω, and adjoin λ many subsets of ℵ _ω which are unions of ω-sequences of those Cohen subsets; then let N be a choiceless submodel generated by equivalence classes of the λ subsets of ℵ _ω modulo an appropriate equivalence relation.     

An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L _p,k, and the finite field F(p ^k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ₁ of the variety V(L _p,k) generated by L _p,k into the variety V(F(p ^k)) generated by F(p ^k) and an interpretation Φ₂ of V(F(p ^k)) into V(L _p,k) such that Φ₂Φ₁(B) = B for every B ε V(L _p,k) and Φ₁Φ₂(R) = R for every R ε V(F(p ^k)).