Countable Cofinality Research Articles

Stationary reflections for uncountable cofinality

It was J. E. Baumgartner who in [1] proved that when a weakly compact cardinal is Lévy-collapsed to ω2 the new ω2 inherits some of the large cardinal properties; e.g. if S is a stationary set of ω-limits in ω2 then for some α < ω2, S ∩ α is stationary in α. Later S. Shelah extended this to the following theorem: if a supercompact cardinal κ is Lévy-collapsed to ω2, then in the resulting model the following holds: if S ⊆ λ is a stationary set of ω-limits and cf(λ) ≥ ω2 then there is an α. < λ such that S ∩ α is stationary in α, i.e. stationary reflection holds for countable cofinality (see [1] and [3]). These theorems are important prototypes of small cardinal compactness theorems; many applications and generalizations can be found in the literature. One might think that these results are true for sets with an uncountable cofinality μ as well, i.e. when an appropriate large cardinal is collapsed to μ++. Though this is true for Baumgartner's theorem, there remains a problem with Shelah's result. The point is that the lemma stating that a stationary set of ω-limits remains stationary after forcing with an ω2-closed partial order may be false in the case of μ-limits in a cardinal of the form λ+ with cf(λ) < μ, as was shown in [8] by Shelah. The problem has recently been solved by Baumgartner, who observed that if a universal box-sequence on the class of those ordinals with cofinality ≤ μ exists, the lemma still holds, and a universal box-sequence of the above type can be added without destroying supercompact cardinals beyond μ.

Blunt and topless end extensions of models of set theory

AbstractLet be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that has a Σn end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If is a well-founded model of ZFC, such that has uncountable cofinality and has a topless Σ3 end extension, then has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in ) highly hyperinaccessible. In §3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σn end extension for each n ∈ ω. The case κ = ω1 is investigated more deeply in §4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω1-like models of ZFC are constructed which have a Σn end extension for all n ∈ ω but have no elementary end extension. ω1-like models of ZFC which have no Σ3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions.

The Baire category theorem and cardinals of countable cofinality

AbstractLet κB be the least cardinal for which the Baire category theorem fails for the real line R. Thus κB is the least κ such that the real line can be covered by κ many nowhere dense sets. It is shown that κB cannot have countable cofinality. On the other hand it is consistent that the corresponding cardinal for 2ω1 be ℵω. Similar questions are considered for the ideal of measure zero sets, other ω1, saturated ideals, and the ideal of zero-dimensional subsets of Rω1.

The Weight of a Pseudocompact (Homogeneous) Space Whose Cardinality Has Countable Cofinality

The Weight of a Pseudocompact (Homogeneous) Space Whose Cardinality Has Countable Cofinality

The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality

Let X be an infinite pseudocompact space. We are interested in restrictions on κ = | X | \kappa = |X| and λ = w ( X ) \lambda = w(X) in addition to the obvious inequalities λ ⩽ 2 κ \lambda \leqslant {2^\kappa } and κ ⩽ 2 λ \kappa \leqslant {2^\lambda } and κ ⩾ 2 ω \kappa \geqslant {2^\omega } , valid for X without isolated points (in particular for homogeneous X). We show that if cf ( κ ) = ω {\text {cf}}(\kappa ) = \omega then λ ⩽ 2 > κ \lambda \leqslant {2^{ > \kappa }} , and even λ ⩽ 2 μ \lambda \leqslant {2^\mu } for some μ > κ \mu > \kappa if X is homogeneous. Under the Singular Cardinals Hypothesis (which is much weaker than the GCH), there are no further restrictions for X without isolated points.