The Relative Consistency of 𝔤 < cf(Sym(ω))

Heike Mildenberger & Saharon Shelah

We give a forcing construction for a model where the groupwise density number 𝔤 is ℵ_{1} and the bounding number, 𝔟 and the cofinality of the symmetric group, cf(Sym(ω)), are both ℵ_{2}.

Simon Thomas asked whether 𝔤 ≠ cf(Sym(ω)) is consistent [?, Question 3.1].

Though the iterands have the countable chain condition and iteration is with finite supports, the construction works so far only for length ℵ_{2} because of the preservation theorems we prove along the way.

Addresses Heike Mildenberger: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel

E-mail: heike@math.huji.ac.il

Saharon Shelah: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel