# Function $f:\kappa\to\alpha$ with small fibers where $\alpha\in\kappa$

Is it consistent in $$\mathsf{ZF}$$ that there is an infinite cardinal $$\kappa$$, cardinals $$\alpha, \beta\in\kappa$$ and a function $$f:\kappa\to \alpha$$ such that for each $$x\in\alpha$$ there is an injective map $$i:f^{-1}(\{x\})\to \beta$$?

• I think in ZFC one can write $\kappa=|\Gamma_f|=|\coprod_{x\in \alpha}f^{-1}(\{x\})|\leq\alpha\beta=\mathrm{max}(\alpha,\beta)<\kappa$? Which point fails in ZF? Nov 10, 2018 at 9:40
• @Qfwfq How do you prove $|\coprod_{x\in \alpha}f^{-1}(\{x\})|\leq\alpha\beta$ in $\mathrm{ZF}$ if you don't have a choice function $x \mapsto i_x \colon f^{-1}(\{x\}) \to \beta$? Nov 10, 2018 at 11:37
• Do $\alpha,\beta,\kappa$ have to be alephs? I guess so, because you are treating them as sets?
– bof
Nov 10, 2018 at 11:42
• @Qfwfq You're welcome. Nov 10, 2018 at 17:29

Consider a (transitive) model of $$\mathrm{ZF}$$ in which $$\omega_1$$ has countable cofinality. Fix a strictly increasing, cofinal sequence $$(\xi_n \mid n < \omega)$$ in $$\omega_1$$. Consider $$f \colon \omega_1 \to \omega, x \mapsto \min \{n < \omega \mid x < \xi_n \}.$$ $$f^{-1}(\{n\})$$ is bounded in $$\omega_1$$ for all $$n \in \omega$$. Hence there is an injection $$i \colon f^{-1}(\{n\}) \to \omega.$$