A uniform ultrafilter over a singular cardinal with a singular character
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A UNIFORM ULTRAFILTER OVER A SINGULAR CARDINAL WITH A SINGULAR CHARACTER M. GITIK School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel e-mail: [email protected] (Received October 15, 2019; revised November 12, 2019; accepted November 13, 2019)
Abstract. We construct a model with a uniform ultrafilter U over a singular strong limit cardinal κ such that κ < cof(Ch(U )) < Ch(U ) < 2κ .
1. Introduction Let U be an ultrafilter over an infinite cardinal κ. We shall say that U is uniform iff |A| = κ whenever A ∈ U . A subset W of U is called a base iff for every A ∈ U there is B ∈ W such that B ⊆∗ A, i.e. |B \ A| < κ. A trivial example is W = U , but we are interested in small bases. The character of U , Ch(U ) is min{|W| | W is a base of U }. We continue here the study of uniform ultrafilters over singular cardinals started by S. Garti and S. Shelah in [2] and continued in [3], [4]. It is easy to construct a uniform ultrafilter U on an infinity cardinal κ such that Ch(U ) = 2κ . If we force 2κ to be singular, then a singular character is obtained. Hence, the interesting problem regarding singular characters is the possibility that Ch(U ) < 2κ and yet singular. Our aim here will be to construct a model with a uniform ultrafilter U over a singular strong limit cardinal κ having a singular character below 2κ . Namely, in our model κ < cof(Ch(U )) < Ch(U ) < 2κ . It answers Question 3.4(ℵ) of [4]. The construction uses extenders and forcing notions associated with them. We will describe this in the next section. The final section will contain the construction itself. The research was partially supported by Israel Science Foundation, Grant no. 1216/18. We are grateful to the referee of the paper for his corrections and suggestions. Key words and phrases: ultrafilter, singular character, forcing. Mathematics Subject Classification: 03E35, 02E55.
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2. Forcing notions and extenders In this section we will briefly describe the relevant forcing notions. Let κ < λ be uncountable cardinals. By an extender E over κ of the length λ we will mean a directed sequence Ea | a ∈ [λ] κ and let it be the trivial forcing otherwise. Now proceed as in Cummings [1, pp. 877–879], only replacing Add(i(μ), i(μ++ )) there by Add(i(μ), i(ℵμ+ )). The final forcing P ∗ S0 will be as desired. Namely, if V1 denotes a resulting generic extension of V then V1 satisfies the following: 1. μ is measurable, 2. 2μ = ℵμ+ , 3. all cardinals (and cofinalities) are preserved, 4. no new sequences of ordinals of the length κ++ are added. Now, E remains an extender over κ of the length λ in V1 , since no new subsets are added to κ. However E is not anymore a (κ, λ)-extender. It is possible to fix this using indestructibility of a strong result from [7], but this will not be needed here. The last item implies that the extender order ≤E is still κ++ -directed. The next stage will be to force over V1 with the extender based Prikry forcing PE with E. We refer to [6, Section
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