More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.     

Whitehead groups may be not free,even assuming CH,I

We prove the consistency with ZFC+G.C.H. of an assertion, which implies several consequences of , which implies their negation. The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research by grant 1110.     

The spectrum problem I: ℵɛ-satuarted models,the main gap

The main goal of this paper is to complete the classification of those first-order theories such that \(I_{N_0 }^a (\lambda ,T) = 2^\lambda \) . We introduce two notions, thedimensional order property anddeepness. Our main theorem asserts that for a superstable theory \(T,I_{N_{\text{0}} }^a (\lambda ,T) = 2^\lambda \) iffT has the dimensional order property or is deep. In a sense made precise in §4 this provides a syntactical characterization of theories with the maximum number of ??-saturated models in each power.     

Less Saturated Ideals

We prove the following:

(1)

If is weakly inaccessible then is not -saturated.

(2)

If is weakly inaccessible and is regular then is not -saturated.

(3)

If is singular then is not -saturated.

Combining this with previous results of Shelah, one obtains the following:

(A)

If then is not -saturated.

(B)

If then is not -saturated.

     

On the strong equality between supercompactness and strong compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension ZFC + GCH in which, (a) (preservation) for regular, if is supercompact', then is supercompact' and so that, (b) (equivalence) for regular, is strongly compact' iff is supercompact', except possibly if is a measurable limit of cardinals which are supercompact.

     

The monadic theory of (ω 2, <) may be complicated

Summary Assume ZFC is consistent then for everyB there is a generic extension of the ground world whereB is recursive in the monadic theory of ₂.The second authro would like to thank the United States — Israel Binational Science Foundation for partially supporting this research. Publ. 411     

Maximal chains inωω and ultrapowers of the integers

Summary Various questions posed by P. Nyikos concerning ultrafilters on and chains in the partial order (, <^*) are answered. The main tool is the oracle chain condition and variations of it.The first author is partially supported by the basic research fund of the Israeli Academy. The second author is partially supported by NSERC and was a guest of Rutgers University while the research on this paper was being done. The authors would also like to thank P. Nyikos for his valuable comments on early versions of this paper. This is number 465 on the first author's list of publications