Cardinal invariants from continuous Ramsey theory - CiteSeerX

Cardinal invariants from continuous Ramsey theory Weinert, Thilo Volker advisor : Prof. Dr. Stefan Geschke

The tale and the goals

Notions & Facts

The topos of this research can be traced back to 1878 when the mathematician Georg Cantor stated his famous continuum problem. He asked whether every uncountable set of reals were already equinumerous to the set of all reals. In modern terms, assuming the axiom of choice, he asked whether or not ℵ1 = c. In 1938 Kurt G¨odel proved the continuum hypothesis to be not refutable in ZFC by showing that it in fact holds true in the inner model of all constructible sets L. Then in 1963 Paul Cohen proved it to be unprovable in ZFC. For this he invented and employed the method of forcing. This method allows to add further sets to a given model of set theory in such a way that what one gets still satisfies all axioms of ZFC. However certain properties of the model may have changed, for example whether or not the continuum hypothesis holds true. The theory of cardinal invariants may be looked upon as a refinement of the continuum problem. After it had been shown that the continuum hypothesis is independent from ZFC the standard axiom system of set theory, it was discovered that one could in many different ways define cardinals combinatorily which have to lie naturally somewhere in between ℵ1 and c, i.e. they are necessarily uncountable and are never larger than the continuum. Yet the precise value of them often depends on the model. The standard technique for proving consistency results in the theory of cardinal invariants consists in the method of iterated forcing. This method is a refinement of Paul Cohen’s original forcing method. It allows adding sets in transfinitely many stages after another. Adding sets to a model in this linearly ordered fashion gives one additional control of the theory of the final model. This is mainly because many sets which were added by this forcing iteration were already added at an intermediate stage so one is still capable to react to that. Problems which arise in the theory of cardinal invariants are often the driving force behind the invention of new techniques for proving independence results. This is to a great extent what makes this field of research so interesting and important. Currently there are two main areas in the topic of cardinal invariants of the continuum. One deals with invariants defined from σ-ideals on the set of all reals, the most important ideals in this context are the ideals of sets of Lebesgue measure zero and the ideal of meager sets. The other deals with invariants which are combinatorially defined in a more direct way and commonly denoted by small gothic letters. The relationships between the cardinal invariants of the first kind are depicted in what is called “Cicho´ n’s diagram ”, for the invariants of the second kind one has “van Douwen’s diagram”. The diagram below combines homogeneity numbers with invariants from both areas which allow to ask nontrivial questions concerning the relationship of the homogeneity numbers to each other and some invariants from van Douwen’s diagram. Their position within Cicho´ n’s diagram is clear, they all have at least the cardinality of cof(null), the largest invariant in it. They tend also to be large in another sense, they can lie at most one cardinal below the continuum. The canonical model in which cardinal invariants are small is the Sacks model. We conjecture that in this model all homogeneity numbers are ℵ1 . However until now this is only known for n < 4. Moreover we conjecture that for all natural numbers n it is consistent that hmn < hmn+1 = c. Solutions of related problems suggest that such models should be easier to construct for c = ℵ2 than for larger values for c. In fact for no n ∈ ω \ 2 it is known whether ℵ1 < hmn < c is consistent. Another open question is how the homogeneity numbers relate to cardinal invariants from van Douwen’s diagram as say i. We have applied for a DFG-grant for this research project in order to answer these questions. However work on this topic has already begun in the spring of this year.

Definition 1 The uniformity number non(I) of a σ-ideal I on a set X is defined as the least cardinal κ such that there is a subset of X of size κ which is not in the ideal, i.e. κ := min{#(Y )|Y ∈ P(X) \ I}. The covering number cov(I) of a σ-ideal I on a set X is defined as the least cardinal κ such that S there is a family of size κ of sets from the ideal which covers the whole set, i.e. κ := min{#(F )|F ⊂ I ∧ F = X}. The cofinality cof(I) of a σ-ideal I on a set X is defined as the least cardinal κ such that there is a family of size κ of sets from the ideal which is cofinal in the ideal, i.e. κ := min{#(C)|C ⊂ I ∧ ∀x(x ∈ I → ∃y(y ∈ C ∧ x ⊂ y))}. Theorem 2 (Andreas Blass) Given an uncountable Polish space X, natural numbers m, n and a Baire colouring b : [X]n −→ m there is a perfect set P ⊂ X such that #(f “[P ]n ) 6 (n − 1)!. The perfect set which witnesses the truth of the theorem above is called weakly homogeneous for b. Definition 3 If n ∈ ω \ 2 the nth homogeneity number hmn is defined as the supremum of all covering numbers of of σ-ideals generated by sets which are weakly homogeneous for a colouring of n-elementic subsets of X. That is hmn := sup{cov(I)|∃m, n, b(m, n < ω ∧ b : [X]n −→ m and I is the σ-ideal generated by the sets weakly homogeneous for b)}. Fact 4 hm2 6 hm3 6 hm4 6 . . . , i.e. ∀n(n ∈ ω \ 2 → hmn 6 hmn+1 ). Theorem 5 (Stefan Geschke) hm+ 2 > c, in the light of fact 4 this means that all homogeneity numbers can be at most one cardinal below the continuum. Theorem 6 (Stefan Geschke) hm2 > cof(null). Definition 7 • a := min{#(F )|F ⊂ P(ω) ∧ ∀x, y ∈ F : #(x ∩ y) < ℵ0 ∧ ∀x ⊂ ω∃y ∈ F : #(x ∩ y) = ℵ0 }. • b := min{#(F )|F ⊂ ωω ∧ ∀f ∈ ωω∃g ∈ F : #({n < ω|f (n) < g(n)}) = ℵ0 }. • d := min{#(F )|F ⊂ ωω ∧ ∀f ∈ ωω∃g ∈ F : #({n < ω|f (n) > g(n)}) < ℵ0 }. T S • i := min{#(F )|∀x, y ∈ [F ]