On the Generic Type of the Free Group

We answer a question raised by Pillay, that is whether the infinite weight of the generic type of the free group is witnessed in $F_ω$. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not $\aleph_1$-homogeneous.

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