Stability and omitting types

We give a complete solution to the following question: when does a superstable theory have a model of powerκ omitting a partial typeq? In particular, for fixedq, if there is such a model of power ?₁ then there is one of power 2^? ₀; and if there is a model omittingq of power ?₁, then there are arbitrarily large ones. For stable theories, a model of power ? _ω ⁺ , omittingq implies one of power 2^? ₀, and this is sharp. Several improvements and some negative results are listed in the introduction.     

Classification theory for non-elementary classes I: The number of uncountable models ofψ ∈L ω_1, ω. Part A

Assuming that 2^Nn < 2^Nn+1 forn < ω, we prove that everyψ ∈L _{ω_1, ω} has many non-isomorphic models of powerN _n for somen>0or has models in all cardinalities. We can conclude that every such Ψ has at least 2 _N 1 non-isomorphic uncountable models. As for the more vague problem of classification, restricting ourselves to the atomic models of some countableT (we can reduce general cases to this) we find a cutting line named “excellent”. Excellent classes are well understood and are parallel to totally transcendental theories, have models in all cardinals, have the amalgamation property, and satisfy the Los conjecture. For non-excellent classes we have a non-structure theorem, e.g., if they have an uncountable model then they have many non-isomorphic ones in someN _n (provided {ie212-7}).     

Iterated forcing and changing cofinalities

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some interations) are preserved, and it includes some forcing which changes the confinality of a regular cardinal >ℵ₁ to ℵ₀. So, using the right iteractions, we can iterate such forcing without collapsing ℵ₁. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.: (1) for everyS ⊑ ℵ₂,S or ℵ₂-S contains a closed copy of ω₁ (2) there is a normal precipitous filterD on (3) for every is regular inL (δ ∩A)} is statonary. The results can be improved to equi-consistency; this will be discussed in a future paper. The author thanks the United States-Israel Binational Science Foundation for supporting the research by grant 1110.     

Exponentiation in power series fields

We prove that for no nontrivial ordered abelian group does the ordered power series field admit an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field , no exponential on is compatible, that is, induces an exponential on through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.

     

Evasion and Prediction II

A subgroup G Z exhibits the Specker phenomenon if every homomorphismG Z maps almost all unit vectors to 0. We give several combinatorialcharacterizations of the cardinal e, the size of the smallestG Z exhibiting the Specker phenomenon. We also prove the consistencyof > e, where is the unbounding number and e the evasionnumber. Our results answer several questions addressed by Blass.     

Coloring finite subsets of uncountable sets

It is consistent for every that and there is a function such that every finite set can be written in at most ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least ways as the union of two sets with the same color.

     

Gross spaces

A Gross space is a vector space of infinite dimension over some field , which is endowed with a symmetric bilinear form and has the property that every infinite dimensional subspace satisfies dim dim. Gross spaces over uncountable fields exist (in certain dimensions). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals , a Gross space in dimension exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.

     

Strongly almost disjoint sets and weakly uniform bases

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of for singular is proved. CECA is used to show that certain ``almost point-' families can be refined to point- families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable -space with a weakly uniform base has a point-countable base.'

     

Constructing Boolean Algebras for cardinal invariants

We construct Boolean Algebras answering some questions of J. Donald Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with 'few automorphisms', with entangled sequences of linear orders, and with semi-ZFC examples of the non-attainment of the spread (and hL, hd).     

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.